Q: According to the fundamental principle of counting, if you have $$a$$ possible outcomes for one event and $$b$$ possible outcomes for another event, how do you find the total number of outcomes for both events?

Multiply $$a$$ and $$b$$: $$a \times b$$

Q: How is $$5! (5 $$factorial) calculated?

$$5 \times 4 \times 3 \times 2 \times 1 = 120$$

Q: You need to create a 4-digit PIN using the 10 digits (0-9), with no repeats allowed. How many different PINs can you create?

$$10 \times 9 \times 8 \times 7 = 5040$$

Q: In a PE class of 30 students, the teacher will award prizes to the top 3 finishers in a race. How many different ways can the top 3 be arranged (order matters)?

$$30 \times 29 \times 28 = 24360$$

Q: At 'A Dip in the Bay', you want to choose a cone to eat. There are 4 types of cones (choose 1), 20 types of ice cream (choose 3), and 5 types of toppings (choose 2). How many different combinations can you create?

$$4 \times \frac{20!}{3! \times 17!} \times \frac{5!}{2! \times 3!}$$

Question 1

According to the fundamental principle of counting, if you have $$a$$ possible outcomes for one event and $$b$$ possible outcomes for another event, how do you find the total number of outcomes for both events?

Accepted Answer

Multiply $$a$$ and $$b$$: $$a 	imes b$$

Question 2

A couple plans on having 4 children. Each child can be either a boy or a girl. How many different girl/boy outcome sequences are possible?

Accepted Answer

$$2^4 = 16$

Question 3

You take an 'Always', 'Sometimes', and 'Never' quiz in Calculus with 8 questions. If you guess on every question, how many different ways can you answer the quiz?

Accepted Answer

$$3^8 = 6561$

Question 4

At McDonald's, there are 31 sandwich options, 3 sizes of fries, and 3 sizes of drinks. How many different meal combinations can you order?

Accepted Answer

$$31 	imes 3 	imes 3 = 279$$

Question 5

How is $$5! (5 $$factorial) calculated?

Accepted Answer

$$5 	imes 4 	imes 3 	imes 2 	imes 1 = 120$$

Question 6

By definition, what are the values of $$0!$$ and $$1!$$?

Accepted Answer

$$0! = 1$$ and $$1! = 1$$

Question 7

A librarian has a pile of 20 books and needs to choose 5 to place in a specific order on the shelf. How many different options does she have?

Accepted Answer

$$\frac{20!}{(20-5)!}$$

Question 8

You need to create a 4-digit PIN using the 10 digits (0-9), with no repeats allowed. How many different PINs can you create?

Accepted Answer

$$10 	imes 9 	imes 8 	imes 7 = 5040$$

Question 9

You need to create a 4-digit PIN using the 10 digits (0-9), with repeats allowed. How many different PINs can you create?

Accepted Answer

$$10^4 = 10000$

Question 10

In a PE class of 30 students, the teacher will award prizes to the top 3 finishers in a race. How many different ways can the top 3 be arranged (order matters)?

Accepted Answer

$$30 	imes 29 	imes 28 = 24360$$

Question 11

If the PE teacher only cares about which 3 students finish in the top 3 (order does NOT matter), how many different groups of 3 can be chosen from 30 students?

Accepted Answer

$$\frac{30!}{3! 	imes 27!}$$

Question 12

How many different juries of 12 can be selected from a pool of 30 people?

Accepted Answer

$$\frac{30!}{12! 	imes 18!}$$

Question 13

At volleyball tryouts, there are 18 girls and only 12 spots on the team. How many possible teams can be formed?

Accepted Answer

$$\frac{18!}{12! 	imes 6!}$$

Question 14

You are on standby for a flight to Alaska. There are 12 people waiting, but only 3 seats available. How many different combinations of people can be chosen for the flight?

Accepted Answer

$$\frac{12!}{3! 	imes 9!}$$

Question 15

At 'A Dip in the Bay', you want to choose a cone to eat. There are 4 types of cones (choose 1), 20 types of ice cream (choose 3), and 5 types of toppings (choose 2). How many different combinations can you create?

Accepted Answer

$$4 	imes \frac{20!}{3! 	imes 17!} 	imes \frac{5!}{2! 	imes 3!}$$

Question 16

Which calculator command would you use to compute the number of permutations of 30 objects taken 3 at a time?

Accepted Answer

nPr

Question 17

Which formula should you use to calculate the number of ways to arrange $$n$$ distinct objects taken $$r at a $$time, when order matters and repeats are NOT allowed?

Accepted Answer

$$\frac{n!}{(n-r)!}$$

Question 18

Which formula should you use to calculate the number of ways to choose $$r$$ objects from $$n$$ distinct objects, when order does NOT matter?

Accepted Answer

$$\frac{n!}{r! (n-r)!}$$

Permutations, Combinations, and the Counting Principle: Study Notes for Probability

Study Guide - Practice Questions

Study Guide - Flashcards