VIEWS: 519 PAGES: 8 CATEGORY: Algebra POSTED ON: 2/24/2010 Public Domain
MASTER in & by NgKL (M.Ed.,B.Sc.Hons.Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH.) A. PERMUTATIONS FACTS: 1. Multiplication Rule: If an event A happens in p ways and event B happens in q ways, then the number of ways that event A happens followed by event B is p x q. 2. Permutations: (a) The number of ways to arrange n unlike objects or events = n! where n! = n x (n – 1) x (n – 2) ……3 x 2 x 1. Example: 5! = 5 x 4 x 3 x 2 x 1 = 120 (b) 0! = 1 (c) The number of ways to arrange n objects which have p objects n! are alike = p! Similarly, if in the n objects have p and q objects which are alike, then the number of ways to arrange the n objects = n! p! q! (d) The number of ways to arrange r objects taken out from n n! unlike objects, n P r = ( n r )! 5P3 5! 5 x 4 x3 x 2 x1 Example: = = = 60 ( 5 3 )! 2 x1 3. In a permutations, the order of arrangement of the objects is important. Example: The arrangement of ABC and CBA are different! 2 Exercise A.1: 1. Calculate the number of ways of 2. Calculate the number of ways of forming a mixed-double team in appointing a male class monitor badminton from 5 male and 4 and a female assistant monitor. female players. 3. Calculate the number of ways to arrange the following words or digits. (a) ADDMATHS (b) HIGHSCHOOL (c) 3, 4, 9, 9 (d) 2, 3, 5, 10, 11, 25, 25 4. Find the number of different arrangement of the following; (a) Five letters from the words (b) Four-digits numbers from the PROBLEMS, without digits 1, 3, 5. 6, 7, 8, 9 if no repetitions. repetition is allowed. (c) Five boys be arranged in a (d) 3 chairs be arranged in a row from a total of 8 boys. row from 9 different chairs. 3 Exercise A.2: Problem Solving I 1. How many 5-digit even numbers 2. How many 4-digit numbers, less can be arrange from digits 1, 2, 5, than 5,000 can be formed from the 8, 9 if no repetition is allowed? digits 3, 4, 5, 6 if no repetition is allowed? 3. How many different arrangements 4. How many different arrangements can be formed from the letters of can be formed from a group of 4 the word TERBILANG if the boys and 3 girls, if arrangements arrangement begin with a vowel? begin with a girl? Exercise A.3: Problem Solving II 1. How many 4-digit numbers can be formed from digits 0 to 9, if the numbers are; (a) odd? (b) greater than 8 000? 2. How many 4-letter word codes can be formed from letters of the word HARMONI if the codes; (a) contain letter A? (b) do not contain any vowel? 4 B. COMBINATIONS FACTS: 1. The number of combinations of r objects n not alike objects is nC r given by nC r n! n( n 1 )( n 2 ).....( n r 1 ) ( n r )! r ! r! 4 C2 4! 4x3x2x1 Example: = 6 ( 4 2 )!2! (2x1)(2x1) C0 1 n 2. Cn 1 n 3. 4. In a combination, the order of arrangement of objects is not important. Example: Arrangement of AB and BA are regarded as one combination. Exercise B.1: 1. Find the number of ways of 2. In how many ways of selecting 3 choosing 4 letters from the word brands of hand-phones from 7 MASTERY. different brands? 3. A computer club has to select 5 4. Four persons have to be chosen committee members form 10 from a group of 4 boys and 3 girls. students. Find how many ways of Find how many ways to select selection? them? 5 Exercise B.2: Problem Solving 1. A Parent-Teacher Association has to select 5 teachers from 7 male and 5 female teachers to be represented in the committee. Determine the numbers of ways to select them to the committee if; (a) 3 males are to be selected. (b) At least 2 males to be selected. 2. Four letters are chosen from the word GLORIES. Calculate the number of selections if (a) the letter G must be selected. (b) only one vowel to be selected. 3. A café serves 5 types of food and 3 types of drink for breakfast. A customer can choose from 1 to 3 types of food or drink for a fixed price. Find the number of different choices the customer can make with that price if; (a) only food are to be chosen. (b) only one type of drink must be chosen. 6 TUTORIAL 1. Diagram 1 shows seven cards. U N I F O R M DIAGRAM 1 A four-letter code is to be formed using four of these cards. Find, (a) the number of different four-letter codes that can be formed, (b) the number of different four-letter codes which end with a consonant. [4 marks] Answer: (a) ………………………………….. (b)…………………………………… 2. A debating team consists of 5 students. These 5 students are chosen from 4 monitors, 2 assistant monitors and 6 prefects. Calculate the number of different ways the team can be formed if (a) there is no restriction, (b) the team contains only 1 monitor and exactly 3 prefects. [4 marks] Answer: (a) ………………………………….. (b)…………………………………… 7 Quadran II Sine positive (180o ) 3. Diagram 2 shows five cards of different letters. M A T H S DIAGRAM 2 (a) Find the number of possible arrangements, in a row, of all the cards. (b) Find the number of these arrangements in which the letters A and H are side by side. [4 marks] Answer: (a) ………………………………….. (b)…………………………………… 4. Diagram 3 shows 5 letters and 3 digits. A B C D E 6 7 8 DIAGRAM 3 A code is to be formed using those letters and digits. The code must consists of 3 letters followed by 2 digits. How many codes can be formed if no letter or digit is repeated in each code? [3 marks] Answer: ….………………………………….. 8