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PERMUTATIONS AND COMBINATIONS A. PERMUTATIONS 1. A nickel and a dime are tossed on a table. In how many ways can they fall? 2. If all questions answered in a true-false quiz of ten questions, how many ways are there of answering the entire quiz? 3. How many numbers of three different digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, 6, 7? 4. If there are 12 milers entered in a race, in how many ways can first, second and third place be awarded? 5. There are five main roads between the cities A and B, and four between B and C. In how many ways can a person drive from A to C and return, going through B on both trips, without driving on the same road twice? 6. How many numbers of at most three different digits can be formed from the integers 1, 2,3, 4, 5, 6? 7. How many numbers of at least three different digits can be formed from the integers 1, 2, 3, 4, 5, 6? 8. A tennis club consists of 12 boys and 9 girls. (a) How many mixed doubles teams (one boy and one girl) are possible? (b) In how many ways can a mixed doubles match be arranged? 9. A freshman student must take a modern language, a natural science, a social science, And English. If there are four possible different modern languages, five natural sciences, three social sciences, but each student must take the same English course, in how many different ways can he select his course of study? 10. In selecting an ace, king, queen, and jack from an ordinary deck of 52 cards, how many ways may we choose if: (a) they must be of different suits? (b) they may or may not be of different suits? (c) they must be of the same suit? (d) they must be in a particular suit? 11. If there are eight outside doors in a dormitory, in how many ways can a student enter one and (a) leave by a different door? (b) leave by any door? 12. A baseball stadium has four entrance gates and nine exits. In how many ways may two men enter together, but leave by different exits? 1 PERMUTATIONS AND COMBINATIONS B. PERMUTATIONS 1. How many numbers of three different digits each can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9? 2. How many numbers of three different digits each less than 700 can be formed from the digits in #1? 3. How many numbers of at most three different digits each can be formed from the digits in #1? 4. In how many ways can a class elect a president, vice-president, secretary, and treasurer from a class of 100 students? 5. In how many ways can four boys and three girls be seated in a row containing seven seats (a) if they may sit anywhere? (b) if the boys and girls must alternate? 6. In how many ways can four boys and four girls be seated in a row containing eight seats (a) if they may sit anywhere? (b) if the boys and girls must alternate? 7. A baseball manager insists on having his best hitter bat fourth and the pitcher bat last. In such circumstances, how many batting orders are possible? 8. In how many ways can eight people be seated in a row of eight seats if two people insist on sitting next to each other? 9. A language teacher wants to keep books of the same language together on his shelf. If he has 12 spaces for 5 French, 4 Italian, and 3 German books, in how many ways can they be placed on his shelf? 10. In how many ways can eight people be seated around a table? 11. In how many ways can eight people be seated around a table if two people insist on sitting next to each other? 12. How many license plates can be made using any two letters for the first two places and any of the numbers 0 through 9 for the last three? 13. Do Problem 12, with the condition that no letter or number be repeated. 14. How many permutations are there of the letters of the word: (a) ALGEBRA (b) COLLEGE 15. How may permutations are there of the letters of the word TENNESSEE? 16. In how many ways can four red beads, five white beads, and three blue beads be arranged in a row? 17. In how many ways can seven different colored beads be made into a bracelet? 18. Show that n+1P5 = (n+1) n Pr-1. 19. Solve the equation nP5 = 20 n P3 for n. 20. Find the value of 5P1 + 5P2 + 5P3 + 5P4 + 5P5. 2 PERMUTATIONS AND COMBINATIONS C. PERMUTATIONS 1. (a) How many ―words‖ of four different letters can be made from the letters A, E, I, O, R, S, T? (b) How many of them begin with a vowel and end with a consonant? (c) In how may do vowels and consonants alternate? 2. (a) How many numbers of five different digits each can be formed from the digits 0, 1, 2, 3, 4, 5, 6? (b) How many of them are even? (c) How many of them are exactly divisible by 5? 3. There are nine chairs in a row. In how many ways can four students be seated in consecutive chairs? SUGGESTION: First find the number of ways of choosing four consecutive chairs. 4. (a) In how many ways can 4 English books and 3 French books be arranged in a row on a shelf? (b) In how many of these will the French books be together? 5. In how many ways can 8 books be arranged on a shelf if 3 particular books are to be together? 6. If there are 6 railroads from Chicago to Minneapolis and 4 railroards from Minneapolis to Seattle, in how many ways can one go by rail from Chicago to Seattle via Minneapolis, and return, without going the same line twice? 7. (a) In how many ways can the letter of the word GEOMETRY be arranged so that vowels and consonants alternate? (b) In how many of these is Y the final letter? 8. A basketball team of 5 regulars and 3 substitutes is passing a ball around a circle. In how many ways can the men be arranged in the circle? 9. How many different bracelets can a child make by stringing 8 beads of different colors all at a time? 10. (a) How many seven-digit numbers can be made from the digits 1, 2, 2, 2, 3, 3, 5? (b) How many of them are odd? 11. (a) How many seven-digit numbers can be formed with the digits 2, 2, 2, 3,3, 4, 5? (b) How many of these are greater than 3, 400, 000? (c) How many are greater than 3, 400, 000 and are divisible by 5? (d) How many are greater than 3, 400, 000 and are even? 12. In how many ways can five men and five girls be seated at a round table so that each girl is between two men? 13. In how many ways can four young couples be seated at a round table so that girls and boys are seated alternately? 14. Four couples are to be seated in a row of 8 seats. How many ways could this be done such that couples are together? 15. How many six people can be arranged on six chairs is two people may NOT sit together? 3 PERMUTATIONS AND COMBINATIONS A. COMBINATIONS 1. Find the value of: (a) 7C4 (b) 10C2 (c) 21C19 2. Find the value of: 8C3 + 8C4 + 8C5 + 8C6 + 8C7 + 8C8 3. In how many ways may a committee of 4 be chosen from a group of 25? 4. From a group of 25 Democrats and 18 Republicans, how many committees consisting of 3 Democrats and 2 Republicans are possible? 5. From the group in Problem 4 ,if one of two specific Democrats is to be chairman, how many committees are possible with the same balance of Democrats and Republicans as previously stated? 6. How many football games are played if each of the nine football teams in a certain conference plays each of the other teams in that conference once? 7. On a college baseball squad, there are three catchers, five pitchers, seven infielders, and seven outfielders. How many different baseball nines can be formed? 8. In how many ways can a bridge hand of 13 cards be chosen from a deck of 52 cards? 9. In how many ways can a person get a bridge hand consisting of only aces or face cards? 10. In how many ways can a person get a bridge hand which consists of cards seven or lower? 11. In how many ways can a person get a bridge hand consisting of two aces ,one king, one queen, three jacks, and the six other cards ten or less? 12. From four red balls, five white balls, and six blue balls, bow many selections consisting of five balls can be made if two are to be red, one white, and two blue? 13. From a group of 15 people, how many committees can be formed consisting of two, tree, or four people? 14. In how many ways may a college president’s wife invite (a) two, (b) three (c) two or more, of eight faculty wives to a tea? 15. How many different sums of money can be formed from a penny, nickel, dime, quarter, and half dollar if at least two coins are used? 16. Solve for n in the equation: n+2C4 = 6 nC2 17. Prove: nCr + nCr-1 = n+1Cr 4 PERMUTATIONS AND COMBINATIONS PERMUTATIONS AND COMBINATIONS FINAL ASSIGNMENT 1. Simplify: a) k 3! k 2 ! 7! r 2 !r b) 6! r 1! 2. Solve for n: a) n! = 20(n – 2)! b) 2n + nP2 = 56 c) n 2 ! 57 8! n 2 ! 16 d) n 1! 30 0 n 1! e) nC2 = nC8 f) 12 Cn+1 = 12Cn+3 g) 4(nP3) = 5(n-1P3) h) C4 = 6(nC2) (n+2) C 44 i) 2n 3 n C2 3 j) 3(nC4) = 5(n-1C5) 3. In five-card poker in how many ways could you be dealt 4 of a kind? 4. In five-card poker in how many ways could you be dealt a full house (3 of one kind and 2 of another kind)? 5. In five-card poker in how many ways could you be dealt 2 pairs and 1 card of a different face value? 6. In five-card poker in how many ways could you be dealt 1 pair and 3 cards of different face values? 5 PERMUTATIONS AND COMBINATIONS 7. If 2 letters are taken from the word BACK and 3 letters are taken from the word FRONT, how many: (a) 5-letter arrangements can be made? (b) 3-letter arrangements can be made? (c) 5-letter arrangements can be made that start with C and end in T? 8. Way Out Motors has 6 different cars and 6 different trucks in stock. In how many ways can a set of 3 cars and 2 trucks be chosen: (a) for a display in the showroom? (b) and arranged in a line in the showroom? (c) and arranged in a line in the showroom so that the cars are to the left of the trucks? 9. How many ways can 11 people be divided into a group of 5 and a group of 6 and then arranged in two circles? 10. How many 2 digit numbers can be made using the digits 1,2,3 if no digit can be repeated? 11. How many 4 digit numbers can be made using the digits 1,2,3,4,5,6,7 if no digit can be repeated? 12. How many ways can 4 letters of the word monkey be arranged? 13. How many ways can 3 letters of the word river be arranged? 14. How many ways can 6 letters of the word geometry be arranged? 15. How many ways can 5 letters of the word January be arranged? 6 ANSWERS for pages 1 - 6: PAGE 1: A. PERMUTATIONS 1. 4 2. 1024 3. 120 4. 1320 5. 240 6. 156 7. 1920 8. (a) 108 (b) 9504 9. 60 10. (a) 24 (b) 256 (c) 4 (d) 1 11. (a) 56 (b) 64 12. 288 PAGE 2: B. PERMUTATIONS 1. 504 2. 336 3. 585 4. 94 109 400 5. (a) 5040 (b) 144 6. (a) 40 320 (b) 1152 7. 5040 8. 10 080 9. 103 680 10. 5040 11. 1440 12. 676 000 13. 468 000 14. (a) 2520 (b) 1260 15. 15 120 16. 27 720 17. 360 18. Proof 19. n = 6 20. 325 PAGE 3: C. PERMUTATIONS 1. (a) 840 (b) 240 (c) 144 2. (a) 2160 (b) 1260 (c) 660 3. 144 4. (a) 5040 (b) 720 5. 4320 6. 360 7 7. 676 8. 5040 9. 2520 10. (a) 420 (b) 240 11. (a) 420 (b) 160 (c) 14 (d) 98 12. 2880 13. 144 14. 384 15. 480 PAGE 4: A. COMBINATIONS 1. (a) 35 (b) 45 (c) 210 2. 219 3. 12 650 4. 351 900 5. 84 456 6. 36 7. 18 375 8. 635 013 559 600 9. 560 10. 2 310 789 600 11. 747 952 128 12. 450 13. 1925 14. (a) 28 (b) 56 (c) 247 15. 26 16. n = 7 17. Proof PAGE 5 & 6: PERMUTATION & COMBINATION FINAL ASSIGNMENT 1. (a) k + 3 (b) 7r2(r + 2)(r + 1) 2. (a) n = 5 (b) n = 7 (c) n = 19 (d) n = 5 (e) n = 10 (f) n = 4 (g) n = 15 (h) n = 7 (i) n = 6 (j) n = 10 3. 624 4. 3744 5. 123 552 6. 4 392 960 7. (a) 7200 (b) 3600 (c) 108 8. (a) 600 (b) 72 000 (c) 7200 9. 1 330 560 10. 6 11. 840 12. 360 13. 33 14. 10 440 15. 1320 8 PERMUTATIONS AND COMBINATIONS BINOMIAL EXPANSION 1. Write the complete expansion for: (a) (a+b)3 (b) (a+b)4 2. Write the complete expansion for (2x - y)4. 3. Write the complete expansion for 2 x F 1I G xJ 3 H K 4. What is the 10th term of (x - 2y)20? Do not simplify. 5. 2 F 2I x Find the term containing x in G J . 5 10 H xK 2 3 Fx 1 I . Find the term with no x’s in G 12 H 2x J 4 6. 2 K 2 7. Find the term containing x14 in (2x - x2)11. 8. Write and simplify the first three terms for each of the following: a. (2x + 1)9 b. (2x2 – x)11 7 3 9. Find the fourth term of 3a 2 a 10. Find the 7th term of (a + b)10 F 2I y 11. Find the 5th term of G J 7 H yK 4 Fx 1 I 12. Find the middle term of G J 12 H 2x K 2 13. Find the 6th term of (2y + x)11 Fa 1 I term of G 9 H 6a J st 14. Find the (r + 1) 3 K 2 15. Find the term containing x20 in (2x - x4)14. 9 PERMUTATIONS AND COMBINATIONS EXTRA PERMS AND COMBS 1. a. How many ways can eight people be seated around a circular table? b. How many ways can they be seated if Bob and Ray insist on sitting next to each other? 2. Five men and five women sit around a circular table, men and women alternating. How many ways can this be done? 3. In both a science classroom and a history classroom there are 12 desks. In the science class students are seated in a circle, and in the history class students are seated in a row. Which classroom has the greater number of seating arrangements? 4. How many ways can four beads of different colours be arranged to form a bracelet? 5. How many ways can three good friends be seated together around a circular table with 10 chairs if Brad refuses to sit beside them and five other people are to be seated? 6. How many four-digit numbers larger than 5600 can be made using the digits 0, 1, 2, 5, 6, 8, 9? 7. Using the numbers 1, 2, 3, 5, 6, 8, 0 (no repetitions): a. how many four-digit numbers are possible? b. how many are divisible by five? c. how many are even? 8. How many numbers less than 700 have no repetition of digits? 9. Using the digits 2, 2, 3, 3, 4, 4, 5: a. how many seven-digit numbers can be found? b. how many are greater than 4 300 000? c. how many are greater than 4 300 000 and divisible by five? 10. How many ways can five men and three women be arranged in a row if there is a man at each end of the row? 11. There are 11 chairs in a row. In how many ways can five people be seated if they sit in consecutive chairs? 12. A book collector has five different books by Dickens, three different plays by Shakespeare, and three different novels by Danielle Steele. She also has a short story by Margaret Laurence. How many ways can they be arranged on a shelf if the books by each author are to be kept together? 10 PERMUTATIONS AND COMBINATIONS 13. a. How many ways can eight people be seated around a circular table if George and Monica insist on sitting together? b. If, in addition, Nicky and Brent refuse to be seated together, how many ways can this be done? 14. How many ways can four boys and two girls sit in a movie theatre row (which contains six seats), if a. one boy must be seated on each end? b. all the boys insist on sitting together? 15. a. How many five-letter ―words‖ are possible using the letters in WINTER? b. How many contain ―I‖ as the second letter? c. How many do not start with an E? 16. If all the letters in the word BARRIER are rearranged, a. find the number of permutations. b. find the number of arrangements beginning with the letter R. c. find the number of arrangements beginning with exactly one R. 17. Eight boys are to be arranged in a row. Two particularly unruly boys are not permitted to sit together nor are they allowed to sit at either end of the row. How many ways is this done? 18. How many four-digit numbers greater than 5364 are possible using the digits 1, 2, 3, 5, 7, 8? 19. A school bus can seat 14 people on each side of the aisle. The boys always sit on the left; the girls always sit on the right. When the bus arrives at a stop, it contains nine boys and seven girls. How many ways can four new boys and five new girls be seated? 20. A group of 15 treasure hunters comes to a clearing in the forest from which there are three exits. It is decided that seven will go left, four go by the middle route, and the rest go to the right. How many ways can this be accomplished? 21. How many different bracelets consisting of six beads can be made from 10 differently colored beads? 22. A tennis club has 10 boys and eight girls as members. a. How many matches are possible with a boy against a girl? b. How many matches are possible with two boys against two girls? c. How many matches are possible with a boy and girl against another boy and girl? 11 PERMUTATIONS AND COMBINATIONS 23. In how many ways can a dog team of five be chosen from 10 huskies and eight retrievers, so that the majority of the dog team would be composed of retrievers? 24. A baseball team is to be formed from a squad of 12 people. Two teams made up of the same nine people are different if at least some of the people are assigned different positions. In how many ways can a team be formed if: a. there are no restrictions? b. only two of the people can pitch and these two cannot play any other position? c. only two of the people can pitch but they can also play any other position? 25. A student council consists of a president and eight other members. A yearbook committee of five is to be selected from this group. a. How many ways can this be done if the president must be on the committee? b. How many ways can this be done if the president is not on the committee? 26. A hockey team has nine forwards and three are needed to form a forward line. a. How many possible forward lines involve their top goal scorer? b. If Mario and Serge are two forwards, how many lines include at least one of them? c. How many lines involve their worst three players? 27. How many different four-letter arrangements can be made from the letters in OCTOBER? 28. How many different four-letter arrangements can be made from the letters in DRIVER? 29. Solve for n in the equation: (n + 2)C4 = 6(nC2). 30. How many 5 card hands can be made if there must be 3 of one face value and 2 other cards with different face values? 12 PERMUTATIONS AND COMBINATIONS EXTRA, EXTRA PERMS AND COMBS Exercise 1 1. A ski resort boasts of having 5 lifts from the chalet at the base of the mountain to the tea house half way up the mountain and 3 lifts from the tea house to the summit. In how many ways can a skier get from the chalet to the summit by taking two lifts? 2. A boy has 4 pairs of pants, 3 shirts, 2 suit jackets, 2 pairs of shoes, and 6 ties. In how many outfits can he appear if all of the articles mentioned are different? 3. In how many orders could a ten volume set of craft encyclopedia set be shelved if each of them is placed right side up with the spine facing outwards? 4. If each one of a set of septuplets wants to attend a different high school, in how many ways can they choose their schools if they live in a city with 11 high schools? 5. Evan Hardy has 4 entrances and 9 exits (the 4 entrances and an additional 5 crash doors). In how many ways can a student enter the school and leave by a different door? 6. How many boy-girl dancing couples could be formed if 85 boys and 102 girls attend a school dance? 7. A student is taking Math A 30, Chemistry 20, History 20, English 20, and French 20. If the school has 5 math teachers, 3 chemistry teachers, 4 history teachers, 8 English teachers, and 2 French teachers, how many teacher assignments could the student receive? 8. A company designs jackets that have three stripes of equal width but different color. The background color of the jackets is different than the color of any of the stripes. How many different color patterns can the company produce if it stocks material in 12 colors? 9. In how many ways can three door prizes, one each of $100, $50, and $25, be given out if 30 people have entered the contest and no one can win more than one prize? 10. How many seating arrangements can be formed by 5 people traveling in a 5-passenger car if only 2 of the 5 people are willing to drive? 11. How many seating arrangements can be formed by 5 people traveling in a 7-passenger van if only 2 of the 5 people are willing to drive? 12. How many signals can be made by raising three flags, one above the other, if there are 6 differently colored flags from which to choose? 13 PERMUTATIONS AND COMBINATIONS 13. How many different batting orders can a baseball coach form if all 9 players must bat? 14. In how many ways can a 10-question true-false test be answered if each of the questions is attempted? 15. In how many ways can a 10-question multiple choice test be answered? Assume that there are four choices for each question, each question is answered, and the same choice is never made on any two consecutive questions (if (b) is chosen for question 1, then it will not be chosen for question 2). 16. How many 3-number combinations might a student have to try in order to open a lock if the numbers from 0 to 39 appear on the dial? Assume that the second number may be the same as the first (since the dial will be rotated in a different direction), but the third number must be at least 5 away from the second. 17. In how many ways can all of the letters of the word HEXAGON be arranged? 18. In how many ways can all of the letters of the word HEXAGON be arranged if the arrangement must start with H and end in N? 19. In how many ways can all of the letters of the word HEXAGON be arranged if the middle letter in the arrangement must be X? 20. In how many ways can all of the letters of the word HEXAGON be arranged if the arrangement must start with a consonant and end in a vowel? 21. In how many ways can all of the letters of the word HEXAGON be arranged if the arrangement must not end in a vowel? 22. In how many ways can all of the letters of the word HEXAGON be arranged if the consonants and vowels must alternate in their positions in the word? 23. How many 7-digit phone numbers can be made if the first digit cannot be a 0? 24. How many 7-digit phone numbers can be made if the first three digits can only be 3’s or 4’s but there are no restrictions on the last four digits? 25. Assume that a Saskatchewan license plate consists of 3 letters followed by 3 digits (the first of which cannot be 0). How many such license plates can be made? 26. Using the license plate description of question 25, how many vehicles might the police have to track down in order to find the getaway vehicle used in a bank robbery if a witness can only remember the first two letters and the last digit of the license plate? 27. Andrew’s mother always makes his lunch for him. She packs one brown-bread sandwich, one white-bread sandwich, one fruit item, and one dessert item. Andrew likes ham, cheese, salami, chicken, turkey, and jam as sandwich fillings. He likes apples, grapes, bananas, and pears, and he likes Oreo cookies, chocolate cake, fudge, butterscotch pudding, and O’Henry clusters for dessert. How many lunches can Andrew have if his mother puts a different filling in each sandwich? 28. How many lunches can Andrew have if his mother uses the same kind of filling for both sandwiches. (question 27)? 14 PERMUTATIONS AND COMBINATIONS 29. The S.R.C. decided to run a ―Guess the Baby Picture‖ contest. Six members brought their baby pictures to school and made a poster placing their names along the left side and their baby pictures, in random order, along the right side. Students entering the contest were to match the name with the correct baby picture. If each entry costs $1.00, how much would a student have to spend in order to guarantee a win? 30. To try and improve her writing ability, Agnes bought a thesaurus calculator. After writing the sentence, ―The big house stood empty.‖, Agnes looked up ―big‖, ―house‖, ―stood‖, and ―empty‖ in her electronic thesaurus and found 5 words that could replace ―big‖, 4 words that could replace ―house‖, 3 words that could replace ―stood‖, and 6 words that could replace ―empty‖. In how many ways might Agnes convey the message of the sentence if she uses any of her replacement words or any of her original words? 31. Slot machines have 3 dials with 20 symbols on each dial as shown in the chart below. (a) In how many different ways can the 3 dials stop? (b) In how many ways can one get 3 bars? (c) In how many ways can one get 3 plums? (d) Which combination of three identical symbols is the most likely to get? (e) Which combination of any three symbols is the most likely to get? (f) Which combination of three symbols can you get in only 1 way? Symbol Dial 1 Dial 2 Dial 3 Bar 2 1 1 Bell 1 8 7 Cherry 2 7 0 Lemon 0 0 5 Orange 8 2 4 Plum 7 2 3 Total 20 20 20 32. S7H 3V8 is a typical Saskatchewan postal code. How many postal codes can be formed that begin with S if any digit may be used in the digit positions and any letter may be used in the other letter positions? 33. In a smaller high school 50 grade 11 students who were taking Math A 30 each designed a questionnaire on an aspect of teen behavior. Their teacher suggested that they conduct their survey by phoning each of the 50 grade 10 students in the school. How many different conversations will have taken place when all of the grade li’s have completed their survey? 34. Three differently colored six-sided dice are rolled. In how many different ways can the dice turn up? 35. A tooney, looney, quarter, dime, nickel, and penny are tossed. In how many different ways might they show heads and/or tails? 36. How many positive two-digit integers can be made from the digits {2,4,5,8 } if digits may not be repeated? 37. How many positive three-digit integers can be made from the digits { 3,4,5,6,7 } if digits may be repeated? 15 PERMUTATIONS AND COMBINATIONS 38. Using the digits { 1,2,5,6,7,9 } and not allowing repetition of digits, how many positive three-digit integers can be made that are: (a) odd? (b) even? (c) larger than 500? (d) divisible by 5? 39. How many positive four-digit integers can be formed: (a) if there are no restrictions? (b) that are divisible by 5? (c) that are even? (d) that are odd? (e) that do not contain the digit 3? (f) that contain two even digits followed by two odd digits (like 4671, 8811, 6097)? 40. Positive integers like 393, 5445, 78987, 444444 are called palindromes--they are unchanged when their digits are written in reverse order. How many five-digit palindromes are there? 41. How many positive integers between 3000 and 4000 can be made with the digits (3,4,6,9) if repetition of digits is allowed? 42. (a) How many positive five-digit integers start with the digit 1? (b) How many positive five-digit integers end with the digit 0? (c) How many positive five-digit integers end with an even digit that is not 0? (d) How many positive even five-digit integers are there? (Hint: look at your answers to (b) and (c).) (e) How many positive odd five-digit integers can be formed so that the odd and even digits alternate? 43. Andrea, Brian, Carol, David, Emmalee, Floyd, and Gloria are to stand in a line for a photograph. If they each face the photographer, in how many ways can they be arranged if: (a) there are no restrictions? (b) the girls and boys must alternate? (c) the boys must be together on the left and the girls must be together on the right? (d) Andrea and Gloria must be standing at either end? (e) Andrea and Brian must stand beside one another? (f) Andrea and Brian must not stand beside one another? (g) the boys must be together? (h) the girls must be together? (i) Floyd must not stand by a girl? (Hint: use cases.) 44. Morse code is a communication system that uses sequences of electronic impulses (dots and/or dashes) to represent characters such as letters, digits, and punctuation marks. For example the letter A is represented by • —, S is represented by • • •, and 2 is represented by • • — • •. How many different characters can be represented by using a sequence of anywhere from 1 to 5 dots and/or dashes? 45. How many positive integers less than 100 000 have the digits 2, 3, and 4 in the order given? (Hint: use cases.) 16 PERMUTATIONS AND COMBINATIONS 46. How many positive even three-digit integers less than 500 can be formed with the digits (1,2,3,4,5,6) if: (a) no digit is repeated? (b) digits can be used no more than twice? (c) digits can be used any number of times? 47. If three different books have yellow covers, three different books have blue covers, and four different books have pink covers, find the number of ways in which the ten books can be arranged on a shelf so that all books of the same color are kept together. Exercise 2 1. In how many ways can all of the letters of the word CANADA be arranged? 2. In how many ways can all of the letters of the word MOOSOMIN be arranged? 3. In how many ways can all of the letters of the word SUPERCALIFRAGILISTICEXPIALIDOCIOUS be arranged? 4. In how many ways can a grandmother distribute 5 dimes, 4 quarters, 3 loonies and 2 toonies to her 14 grandchildren so that each grandchild receives exactly one coin? 5. In how many ways can 6 identical math texts and 3 identical physics texts be arranged on a shelf? Assume the books are arranged right-side-up with the spine facing outwards. 6. In how many ways can 4 blue, 4 gold, 4 green, 4 red and 4 white bulbs, be arranged on a straight string of lights that will hold 20 bulbs if all the bulbs are identical except for color? 7. In how many ways can 3 yellow beads, 4 blue beads and 2 black beads be strung along a straight wire if all of the beads are identical except for color? 8. Along how many different paths can one walk a total of 20 blocks by going a total of 5 blocks north, 5 blocks south, 5 blocks east, and 5 blocks west in any order? 9. How many 5 digit numbers can be made using each of the digits (1,1,1,3,31 exactly once? 10. How many 6 digit numbers can be made using each of the digits {8,8,9,5,5,6} exactly once? 11. How many more arrangements are there from all of the letters of the word MARMALADE than there are from all of the letters of the word MARINADE? 12. In how many different orders can a team win 4 games and lose 3 games in a seven-game round-robin tournament? 13. In how many different orders can the Blades win 4 games and lose 3 games in a best-of- seven playoff series? (Hint: What is the outcome of the seventh game?) 14. In how many ways can an expression equivalent to 5x 4y5z3 be written if I is the only exponent that can be used? One equivalent expression is x•y•z•5•z•y•y•x•x•x•z•y•y. 17 PERMUTATIONS AND COMBINATIONS 15. In a cola taste test, 5 glasses of Pepsi and 5 glasses of Coke were placed at random in a line. In how many orders can the 10 glasses of cola be arranged? 16. The melody of the first 8 bars of the well known hymn, ―Amazing Grace‖ is shown below. How many different melodies could be made by rearranging the notes if only the pitches and not the time values are considered? (Sorry about the missing time signature and the missing treble clef.) 17. In how many different color orders can a standard deck of 52 playing cards be stacked? 18. In how many ways can all of the letters of the word BOOKKEEPER be arranged if: (a) the arrangement must start with B? (b) the arrangement must start with E? (c) the arrangement must start with K and end in 0? (d) the vowels must all be together? (e) the vowels must all be together and the consonants must all be together? (f) the middle two letters must be B and R? 19. Using all of the digits { 1,1,1,2,2,3,3), how many positive seven-digit integers can be written that: (a) have 3 as the middle digit? (b) that begin and end in 3? (c) that begin and end in 1? (d) that have 3 consecutive l’s? (e) are even? (f) are odd? 20. In how many different orders, with respect to wins and losses, can the Regina Pats win a best of seven game playoff series? Some examples are WWWW, WLWWW, LWWWLW, WLWLWLW. Exercise 3 1. In how many ways can 7 children join hands to form a circle if they each face the center? 2. In how many ways can 8 different keys be arranged on a key ring? 3. In how many ways can 5 different keys be arranged on a key ring? 4. In how many ways can 6 children be seated at a round table? 18 PERMUTATIONS AND COMBINATIONS 5. In how many ways can 9 adults be seated at a round table? 6. In how many ways can 7 different charms be arranged on a circular charm bracelet? 7. In how many ways can 6 differently colored beads be arranged on a necklace? 8. In how many ways can 3 boys and 3 girls be seated at a round table if the boys and girls must alternate? 9. In how many ways can 6 men and 6 women be seated at a round table if the men and women must alternate? 10. In how many ways can 4 husband-wife couples be arranged at a circular table if: (a) anyone can sit anywhere? (b) men and women alternate? (c) each man sits beside his wife? (d) each man sits opposite his wife? (e) the men are all together? (f) one of the couples, say Al and Brielle, must sit beside one another? (g) one of the couples, say Al and Brielle, must not sit beside one another? (h) each man sits beside his wife and men and women alternate? (i) each person sits next to a man and a woman? 11. In how many different orders can 3 identical cans of Pepsi and 3 identical cans of Coke be arranged in a circle? It is tempting to try to solve the problem by thinking that since 6 objects are being arranged in a circle they can be arranged in 5! ways, but since there are 3 identical cans of Pepsi and 3 identical cans of Coke, we must reduce our count of 5! by 5! (3 !)(3!). Thus one might think the number of arrangements is Evaluate this 3! 3! expression and indicate why that can’t be correct. Determine the number of possible arrangements by actually drawing them out--there are very few. 12. In how many ways can Pallavi, Tarini, Nathan, Leah, and Ryan be arranged at a circular table if Pallavi must not sit beside Tarini and Nathan must not sit beside Leah? (Hint: Break the problem into cases by using Pallavi as a point of reference and seating Nathan in each of the other chairs.) 13. Ken, Don, Andy, Pam, Lana, and Jen arrange to see a movie. Don would really like to get to know Pam better and hopes that he’ll either sit beside her in the movie or at the restaurant after the movie. If the group is seated randomly in a straight line at the movie and is seated randomly at a circular table at the restaurant, where does Don stand a better chance of being beside Pam, at the restaurant or at the movie? What are his chances in both places? 19 PERMUTATIONS AND COMBINATIONS Exercise 4 1. How many combinations can be formed from the letters of the word HELP taking two at a time? Write them out. 2. How many combinations can be formed from the letters of the word HEXAGON taking three at time? 3. Six points lie on the circumference of a circle. How many triangles can be formed by connecting any three of these points? 4. How many quadrilaterals can be formed by connecting any four points on the circumference of the circle described in question 3? 5. In a history exam a student is asked to select any 5 paragraph questions from among 8 paragraph questions presented. How many different selections can the student make if the order in which the questions is done is not important? 6. A pizza parlor encourages its customers to build their own pizza. If you must choose 5 toppings from among the 15 toppings on the menu, and if the order in which the toppings are placed on the pizza is not important to you, how many different pizzas would you have to order until you had sampled every possible pizza that could be assembled? 7. Lindsey decided to send Christmas cards to 20 friends this year. If she completes 5 cards each day for a period of 4 days, how many different sets of friends does she have to choose from on her first day? 8. A random sample of 10 students is to be chosen from a class of 30 students in order to complete a questionnaire. How many different samples can be chosen? 9. On the opening day of school it is customary for each staff member to greet each of the other staff members with a handshake. How many handshakes will take place if there are 51 members on staff? 10. In how many ways could you select 6 numbers and not match any of the winning numbers while playing Lotto 649? For example the winning numbers might be {6, 15, 22, 24, 30, 411 and you may have selected {7, 14, 21, 28, 35, 42) so none of your numbers match. 11. In the lottery game Prairie Banco, participants can select 11 integers from the integers 1 to 80. How many different selections of 11 integers can be made? 12. Solve for n if nC2 = 100C98. 13. How many different selections of 2 eggs can be made from a carton containing one dozen eggs? 14. If the carton in question 13 contains 4 rotten eggs and 8 good eggs, in how many ways can the selection of 2 eggs be made so that (a) no rotten eggs are selected? (b) no good eggs are selected? 20 PERMUTATIONS AND COMBINATIONS 15. From a class of 20 girls and 14 boys a group of 5 students must be selected to act as guides for the annual open house tours given to grade 8 students. How many different selections of 5 guides can be made if: (a) there are no restrictions? (b) no boys are chosen? (c) no girls are chosen? (d) both boys and girls must be represented? (e) David and Laura must be among those chosen? (f) David and Laura must not be among those chosen? 16. There are 26 red cards and 26 black cards in a standard deck of 52 cards. The red cards are divided into two suits (hearts and diamonds) and the black cards are divided into two suits (clubs and spades). The 13 cards within each suit are known as the 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, king, queen, and ace of that particular suit. In the game of 5-card poker, each player is dealt 5 cards. How many 5-card hands can be dealt such that: (a) any 5 cards might show? (b) all cards are red? (c) all cards are black? (d) both colors of cards are present? (e) there are no hearts? (f) there are only hearts? (g) there are no 2’s or 3’s? (h) there are no face cards? (jacks, kings, and queens are the face cards) (i) there are only face cards? (j) the hand contains all four aces? 17. If you have one of each of a penny, nickel, dime, quarter, looney, and tooney, how many different amounts can you leave as a tip for a waiter if you leave at least one coin? (Hint: Use cases.) 18. In the lottery game SUPER 7 a player must select a set of 7 integers from the integers Ito 47. As in Lotto 649 if there is no winner on a particular game day, the jackpot builds. Explain why SUPER 7 usually has a larger prize than Lotto 649. Exercise 5 1. A small class consists of 5 boys and 7 girls. How many groups can be selected from this class that consist of: (a) any 3 students? (b) any 8 students? (c) 5 girls? (d) 3 boys? (e) 2 boys and 2 girls? (f) 4 boys and 4 girls? (g) 3 students, the majority of whom must be girls? (Hint: Use cases.) 21 PERMUTATIONS AND COMBINATIONS 2. A box contains 4 red marbles, 3 blue marbles, and 5 yellow marbles. In how many ways can one select a set of 5 marbles from the box such that: (a) 2 are red and 3 are blue? (b) 3 are red and 2 are yellow? (c) 2 are red, 1 is blue, and 2 are yellow? (d) none of them are red? (e) none of them are blue? (f) all are yellow? 3. How many 10-card hands can be dealt from a standard deck of 52 cards that consist of: (a) any 10 cards? (b) 7 red cards and 3 black cards? (c) all black cards? (d) all clubs? (e) no clubs? (f) 4 aces, 4 kings, and 2 queens? 4. How many parallelograms of any size are formed by the intersecting parallel lines shown in the figure below? (Hint: It takes a set of 2 horizontal lines and a set of 2 slanted lines to form a parallelogram.) 5. How many 10-card hands can be dealt from a standard deck of 52 cards if: (a) 8 are red and 2 are black? (b) 3 are hearts, 3 are clubs, 2 are diamonds and 2 are spades? (c) exactly 4 of them are face cards? (d) 4 of the cards are aces, 2 are queens, 3 are kings, and 1 is a jack? 6. In how many ways can a committee of 3 students be chosen from a class of 8 boys and 6 girls, so that the girls will form a majority? (Hint: There are 2 cases.) 7. From a class of 9 boys and 10 girls a committee of 3 boys and 3 girls is to be chosen. In how many ways can this be done if: (a) there are no restrictions? (b) Allan must be on the committee but Lindsey must not be on the committee? (c) Allan and Lindsey are either both on the committee or neither of them is on the committee? 8. Pallavi has 3 compact discs and Tarini has 9 compact discs. In how many ways can they exchange discs if each keeps her original number of discs? (Hint: Use cases--they could exchange 1 or 2 or 3 discs.) 22 PERMUTATIONS AND COMBINATIONS 9. Hockey coaches will sometimes juggle lines hoping to find the combination of players that works best together. If there are 9 forwards, 6 defencemen, and 2 goalies on the team, how many: (a) different sets of 3 forwards, 2 defencemen and 1 goalie can be selected by the coach to start the game? (b) different line-ups, according to the specific position played, can be chosen to start the game? (The forward positions are left-wing, center, and right-wing. The defensive positions are left-defense and right-defense. Obviously there is only 1 goalie position. Assume that all forwards are capable of playing any forward position and all defensemen are capable of playing either defensive position.) 10. Using 3 letters from the word ARTICLE and 3 letters from the word SHOW, how many arrangements of six different letters are possible? 11. Using letters from the word MANUSCRIPT, how many 4-letter arrangements consisting of 3 different consonants and 1 vowel can be formed if no letter is repeated? 12. Repeat question 11 if the arrangement must start with N. 13. How many positive 4-digit integers can be made if 2 different digits are selected from { 1, 2, 3, 4) and 2 different digits are selected from {5, 6, 7, 8, 9)? 14. How many positive 5-digit integers can be made if 3 different digits are selected from { 1, 2, 3, 4) and 2 different digits are selected from { 5, 6, 7, 8, 9)? 15. How many positive odd integers of 5 different digits can be formed from the digits (1, 2, 3, 4, 5, 6, 7, 8, 9 } if each integer is to be formed using 3 odd digits and 2 even digits? 16. In bridge a player is dealt 13 cards. (a) How many different 13-card hands are there? (b) The most common 1 3-card hand has 4 cards from one suit, 4 cards from a second suit, 3 cards from a third suit, and 2 cards from the last suit. How many such hands can be dealt? (c) How many 13-card hands can be dealt so that 5 cards are from one suit, 3 cards are from a second suit, 3 cards are from a third suit, and 2 cards are from the last suit? (d) How many 13-card hands have exactly 8 of the cards from the same suit? Review Exercises 1. In how many ways can 4 different books be arranged on a shelf? 2. In how many ways can 5 people be seated around a circular table? 3. A class contains 15 boys and 12 girls. How many different boy-girl dates are possible within the class? 4. In bow many ways can 5 boys line up for a photograph? 23 PERMUTATIONS AND COMBINATIONS 5. From a group of 10 students, how many different committees of 4 students can be formed? 6. In how many ways can 5 keys be arranged on a circular key ring? 7. In how many ways can all of the letters of the word FREEZE be arranged? 8. How many committees of 3 students can be formed from a class of 30 students? 9. How many 3-letter arrangements can be made by using letters from the word CLOUDY if repetition is not allowed? 10. Repeat question 9 if repetition of letters is allowed. 11. In how many ways can 3 identical math texts and 4 identical psychology texts be arranged on a shelf? 12. In how many ways can 15 different charms be arranged on a circular bracelet? 13. A committee of 1 girl and 2 boys is to be selected from a class of 5 girls and 10 boys. How many different committees are possible? 14. How many sets of 3 red cards can be drawn from a standard deck of cards? 15. A restaurant boasts a menu containing 20 different kinds of pizza. If a group of students decide to order 4 pizzas, all different, how many different combinations are possible? 16. Using the digits { 0, 1, 2, 3, 41 and allowing repetition of digits, how many positive: (a) 4-digit integers can be formed? (b) odd 4-digit integers can be formed? (c) 4-digit integers ending in 0 can be formed? (d) 4-digit integers greater than 2000 can be formed? (e) odd 4-digit integers greater than 2000 can be formed? 17. Repeat question 16 assuming repetition of digits is not allowed. 18. (a) In how many ways can 7 people be lined up in a row for a photograph? (b) In how many ways can the 7 people be lined up if one of them, say Carmen, has to be in the middle? (c) How many lineups of the 7 people have Carmen in the middle, and two other people, Al and Brian, always taking the end positions? 19. How many positive 2-digit integers can be formed using the digits { 1, 2, 3) if repetition of digits is not allowed? Write them out. 20. How many positive 4-digit integers can be formed using the digits (1, 2, 3,41 if: (a) repetition of digits is not allowed? (b) repetition of digits is allowed? 21. How many positive 4-digit integers can be formed using the digits { 0, 1, 2, 3} if: (a) repetition of digits is not allowed? (b) repetition of digits is allowed? 24 PERMUTATIONS AND COMBINATIONS 22. Using the digits 0 through 9, how many 5-digit integers can be formed if the digits of the integer must alternate from even to odd or from odd to even as in 98 561 or 65 252? Assume that repetition of digits is allowed. 23. Answer each of the following questions in relation to the four boys A, B, C, and D and the four girls E, F, G, and H. In how many ways can they: (a) stand in a line if D and E must stand beside one another? (b) be seated about a circular table if D and E must be beside one another? (c) be seated about a circular table if the boys must be clustered together and the girls must be clustered together? (d) be seated about a circular table if a boy must have a girl directly across the table from himself? (e) be lined up for a photograph if no two people of the same sex can stand beside one another? (f) line up in two rows for a photograph if the girls must stand in front of the boys? (g) form a 4-member committee consisting of 2 boys and 2 girls? (h) form a 4-member committee consisting of 2 boys and 2 girls if D and E are not allowed to both serve on the same committee? (Use cases.) 24. (a) How many different committees of 4 students can be formed from 8 students? (b) How many of those in part (a) contain one particular student A? (c) How many committees in part (a) include A and exclude B? (d) How many committees in part (a) include both A and B? (e) How many committees in part (a) exclude both A and B? 25. (a) Suppose that a penny is tossed 5 times and that the succession of results is recorded by using H’s for heads and T’s for tails. How many different sequences of results of the 5 tosses are possible? (b) In how many of these sequences is a head showing on both the first and the last toss? 26. A small class consists of 3 boys and 5 girls. (a) In how many ways can they all be seated in a row if both end seats are occupied by boys? (b) In how many ways can a committee of 3 be chosen from this class? (c) How many of the committees in part (b) will contain 1 boy and 2 girls? (d) How many of the committees in part (b) will contain 3 boys? (e) How many of the committees in part (b) will contain 3 girls? 27. Out of a group of 10 boys and 7 girls, 3 boys and 2 girls are to be selected to represent a school. In how many ways can the selection be made? 28. Shannon has 3 boxes, one red, one white, and one blue. She has 9 different objects. In how many ways can she put 3 objects in each box? 29. If the 3 boxes in question 28 are indistinguishable, then it makes no difference in which box Shannon puts a particular set of 3 objects. In how many ways can she now put 3 objects in each box? 30. One section of each of English 20, Mathematics A 30, and Chemistry 20 will be offered in period 1, period 2, and period 3. In how many ways can Emmalee’s schedule be arranged so that she takes each one of these classes? 25 PERMUTATIONS AND COMBINATIONS 31. At each of the times 7:00 p.m., 7:30 p.m., 8:00 p.m., and 8:30 p.m. there are always 3 varieties of TV shows to watch on the three channels available: a western, a mystery, and a comedy. If no program scheduled on one channel ever shows up on another channel, in how many ways can one watch four consecutive TV shows and see: (a) any variety of show? (b) no westerns? (c) at least one western? (Hint: Use your answers from (a) and (b).) (d) only comedies? 32. From a standard deck of cards, how many 10-card hands can be dealt that contain: (a) 6 red cards and 4 black cards? (b) exactly 3 aces? (c) 3 hearts, 2 spades, 2 clubs, and 3 diamonds? 33. Using the digits {0, 1, 2, 3, 4, 5}, and not allowing repetition, how many positive: (a) 3-digit integers can be formed? (b) odd 3-digit integers can be formed? (c) even 3-digit integers can be formed? (d) 3-digit integers can be formed that are larger than 200 but smaller than 500? (e) odd 3- digit integers can be formed that are larger than 200 but smaller than 500? 34. A class contains 10 girls and 10 boys. How many 3-person committees can be formed that will give the boys a majority? 35. How many 13-card hands can be dealt from a standard deck of cards that contain: (a) 8 red cards and 5 black cards? (b) at least 10 red cards? (Hint: Use 4 cases.) (c) exactly 2 clubs and exactly 3 diamonds? (d) exactly 3 sevens and exactly 2 queens? 36. In how many ways can all of the letters in the word ISOSCELES be arranged if arrangement must: (a) start with E and end in E? (b) have S for the middle letter? (c) have the vowels together? (d) have the consonants together? (e) have the consonants and vowels alternate? 26