The Counting Principle
Practice
1. The letters A, B, C, and D are used to form four-letter passwords for entering a computer file. How many passwords
are possible if letters can be repeated any number of times?
2. How many ways can the first five letters of the alphabet be arranged if each is used only once?
3. A restaurant serves 5 main dishes, 3 salads, and 4 desserts. How many different meals could be ordered if each has a
main dish, salad & dessert?
4. How many different ways can 4 different books be arranged on the shelf?
5. How many different 5-digit even numbers can be formed using the digits 4, 6, 7, 2, 8 if digits can be repeated any
number of times?
6. How many 4-digit positive integers are there?
7. How many license plate numbers consisting of three letters followed by three numbers are possible when repetition is
allowed?
8. How many combinations are possible using the information in problem 7 if no repetition is allowed?
9. A briefcase lock has 3 rotating cylinders, each containing 10 digits. How many numerical codes are possible?
10. A golf club manufacturer makes irons with 7 different shaft lengths, 3 different grips, 5 different lies, and 2 different
club head materials. How many different combinations are offered?
11. There are five different routes that a commuter can take from her home to the office. In how many ways can she make
a round trip if she uses a different route coming than going?
12. In how many ways can the 4 call letters of a radio station be arranged if the first letter must be W or K and no letters
repeat?
13. How many 7-digit phone numbers can be formed if the first digit cannot be 0 or 1?
14. How many 7-digit phone numbers can be formed if the first digit cannot be 0 or 1 and if no digit can be repeated?
Permutations
Practice
1. How many ways can 4 charms be arranged on a bracelet that has no clasp?
2. How many different 9-player batting orders can be chosen form a baseball team of 16?
3. In how many ways can the four “legs” of a relay race be assigned from a team of nine runners?
4. In how many ways can 6 students be seated at a table with 4 numbered chairs?
5. In how many ways can a class of 15 students choose a new president, vice president & secretary?
6. In how many ways can the letters of CANADA be arranged?
7. In how many ways can the letters of ANNUALLY be arranged?
8. In how many ways can the letters of MEMBERS be arranged?
Combinations
Practice
1. There are 15 different books. How many groups of 6 books can be selected?
2. How many tennis teams of 6 players can be formed from 14 players without regard to position played?
3. From a standard deck of 52 cards, how many ways can 5 cards be drawn?
4. How many 4-person bobsled teams can be chosen from a group of 9 athletes?
5. From a dessert cart in a fine restaurant, customers are allowed to pick 3 desserts from the 10 that are displayed. How
many combinations are possible?
6. A salad bar offers eight choices of toppings for lettuce. In how many ways can you choose four toppings?
7. In how many ways can 5 apples be chosen from a case of 15 apples?
Basic Probability
Practice
One bag of candy gummy fish contains 15 red fish, 10 yellow fish, and 6 green fish. Find the probability of each
selection, then find the odds of each selection:
1. Picking a red gummy fish 2. Not picking a yellow gummy fish
3. Picking a green gummy fish 4. Not picking a red gummy fish
You have a standard deck of cards (no jokers). Find the probability of each selection, then find the odds:
5. Drawing a red card 6. Drawing a face card
7. Drawing a black 4 8. Drawing an even red card
9. In a bag there are 5 math questions and 4 science questions. Ardie picks a question from the bag. What are the odds
of not picking a science question?
10. What are the odds that a person chosen at random got a passing grade on an algebra test if the scores were 3 A’s, 4
B’s, 10 C’s, 2 D’s, and 2 F’s?
A jar contains 30 red marbles, 50 blue marbles, and 20 white marbles. You pick one marble form the jar at random.
Find each probability.
11. P(red) 12. P(blue) 13. P(not white) 13. P(red or white)
Suppose you roll a number cube (die). Find each theoretical probability.
14. P(5) 15. P(5 or less) 16. P(8) 17. P(an even #)
Multiplying Probabilities
Practice
1. What is the probability of drawing two cards showing odd numbers from a set of cards that show the first 20 counting
numbers if the first card is not replaced before the second is chosen?
2. A jar contains 7 lemon jawbreakers, 3 cherry jawbreakers, and 8 rainbow jawbreakers. What is the probability of
selecting 2 lemon jawbreakers in succession providing the jawbreaker drawn first is then replaced before the second is
drawn?
There are 3 nickels, 2 dimes, and 5 quarters in a purse. Three coins are selected in succession at random.
3. Find the probability of selecting 1 nickel, 1 dime, and 1 quarter in that order without replacement.
4. Find the probability of selecting 1 nickel, 1 dime, and 1 quarter in any order without replacement.
5. Find the probability of selecting 1 nickel, 1 dime, and 1 quarter in that order with replacement.
From a standard deck of 52 cards, 2 cards are selected. What is the probability that the following occurs?
6. 2 black cards; selection without replacement
7. 2 black cards; selection with replacement
8. 1 red card and 1 spade in any order; selection without replacement
9. 1 black card and 1 face card in that order; selection without replacement
A red, a green, and a yellow die are tossed. What is the probability that the following occurs?
10. All 3 dice show 4
11. None of the 3 dice shows 4
12. The red die shows an even number and the other 2 dice show different odd numbers
13. All 3 dice show the same number
Adding Probabilities
Practice
1. A bag contains 35 dyed eggs: 15 yellow, 12 green, and 8 red. What is the probability of selecting a green or a red
egg?
2. A card is selected from a deck of 52 cards. What is the probability of selecting a 7 or a black card?
3. The letters of the alphabet are placed in a bag. What is the probability of selecting a vowel or the letters QUIZ?
4. A card is drawn from a standard deck of 52 cards. What is the probability of selecting a 3, an ace, or a black card?
5. A drawer contains pairs of socks: 4 blue, 3 black, 3 plaid. What is the probability that you will pick a blue or plaid
pair of socks?
6. The letters from the words OAK MOUNTAIN HIGH SCHOOL are placed on cards and put in a box. What is the
probability of drawing a vowel or the letter L?
7. The letters from the words OAK MOUNTAIN HIGH SCHOOL are placed on cards and put in a box. What is the
probability of drawing a vowel or the letters EAGLES?