Q: A food manufacturer samples 400 corn kernels, 3 of which are toxic. If 4 kernels are selected at random, what is the probability that exactly one is toxic?

$$\frac{C(3, 1) \times C(397, 3)}{C(400, 4)}$$

Q: In the Powerball lottery, a player chooses 5 numbers from 1 to 69 and a Powerball from 1 to 26. What is the probability of winning the jackpot with a single ticket?

$$\frac{1}{C(69, 5) \times 26}$$

Question 1

A university assigns unique four-digit codes to students, using the digits 0–9, with no digit repeated in a code. How many different codes are possible?

Accepted Answer

$$P(10, 4) = \frac{10!}{(10-4)!}$$

Question 2

A committee of 3 is to be formed from 20 employees. In how many ways can this committee be selected if the order of selection does not matter?

Accepted Answer

$$C(20, 3) = \frac{20!}{3!17!}$$

Question 3

A building contractor is arranging 6 one-story, 4 two-story, and 2 split-level houses in a row. How many distinguishable arrangements are possible?

Accepted Answer

$$\frac{12!}{6!4!2!}$$

Question 4

A student advisory board of 17 members must select a chair, secretary, and webmaster. What is the probability that three specific members are chosen for these positions, in any order?

Accepted Answer

$$\frac{1}{P(17, 3)}$$

Question 5

A psychologist presents 8 activities to a subject and asks them to pick a first, second, and third activity. How many possible ordered selections are there?

Accepted Answer

$$P(8, 3) = \frac{8!}{5!}$$

Question 6

A standard deck of 52 cards is used. What is the probability of being dealt 5 diamonds in a 5-card hand?

Accepted Answer

$$\frac{C(13, 5)}{C(52, 5)}$$

Question 7

A food manufacturer samples 400 corn kernels, 3 of which are toxic. If 4 kernels are selected at random, what is the probability that exactly one is toxic?

Accepted Answer

$$\frac{C(3, 1) 	imes C(397, 3)}{C(400, 4)}$$

Question 8

In the Powerball lottery, a player chooses 5 numbers from 1 to 69 and a Powerball from 1 to 26. What is the probability of winning the jackpot with a single ticket?

Accepted Answer

$$\frac{1}{C(69, 5) 	imes 26}$$

Question 9

A jury consists of 5 men and 7 women. Three members are selected at random for an interview. What is the probability that all three are men?

Accepted Answer

$$\frac{C(5, 3)}{C(12, 3)}$$

Question 10

Which of the following situations requires the use of combinations rather than permutations?

Accepted Answer

Choosing 5 students from a class to form a committee

Question 11

A company board has 12 members. How many ways can the positions of president, vice president, secretary, and treasurer be assigned?

Accepted Answer

$$P(12, 4) = \frac{12!}{8!}$$

Question 12

A state park has 5 beaches and must select 3 to build new restrooms. How many different groups of beaches can be selected?

Accepted Answer

$$C(5, 3) = \frac{5!}{3!2!}$$

Question 13

Which formula gives the number of distinguishable permutations of the letters in the word 'BALLOON'?

Accepted Answer

$$\frac{7!}{2!2!}$$

Question 14

A password consists of 3 letters followed by 2 digits (0–9), with no repeats. How many different passwords are possible?

Accepted Answer

$$P(26, 3) 	imes P(10, 2)$$

Question 15

Which principle should you use to determine the number of ways to select a president and a vice president from a club of 15 people?

Accepted Answer

Permutation

Question 16

A contractor wants to plant 6 oak, 9 maple, and 5 poplar trees in a row. How many distinguishable arrangements are possible?

Accepted Answer

$$\frac{20!}{6!9!5!}$$

Question 17

A race has 33 cars. In how many ways can the top 3 finishers be arranged?

Accepted Answer

$$P(33, 3) = \frac{33!}{30!}$$

Question 18

Which of the following best describes the difference between a permutation and a combination?

Accepted Answer

A permutation considers order, while a combination does not.

Question 19

A student must answer 5 out of 10 essay questions on an exam. In how many ways can the student choose the questions?

Accepted Answer

$$C(10, 5) = \frac{10!}{5!5!}$$

Question 20

A restaurant offers 3 appetizers, 5 main courses, and 4 desserts. How many different meals (one of each) can a customer order?

Accepted Answer

$$3 	imes 5 	imes 4 = 60$$

Study Guide: Permutations, Combinations, and Applications of Counting Principles in Probability

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