Q: A license plate consists of 2 letters (A-Z) followed by 3 digits (0-9). How many unique license plates are possible?

$$26^2$$ \times $$10^3 = 676$$,000$$

Q: A club has 13 members. How many ways can they elect a president, vice president, and treasurer (all different people)?

$$13 \times 12 \times 11 = 1,716$$

Question 1

A restaurant offers 5 types of bread, 4 types of meat, and 3 types of cheese for sandwiches. If you must choose one of each, how many different sandwiches can you make?

Accepted Answer

$$5 	imes 4 	imes 3 = 60$$

Question 2

How many possible outcomes are there if you roll 4 standard six-sided dice?

Accepted Answer

$$6^4 = 1296$

Question 3

A security system uses a 6-digit code where each digit can be any number from 0 to 9, but the first digit cannot be 0. How many possible codes are there?

Accepted Answer

$$9 	imes 10^5$ = 900,000$$

Question 4

How many different ways can 3 students be chosen from a class of 10 to form a committee, if the order in which they are chosen does NOT matter?

Accepted Answer

$$\dfrac{10!}{3!\,7!} = 120$$

Question 5

A teacher wants to assign 1st, 2nd, and 3rd place awards to 8 students. How many different ways can the awards be assigned?

Accepted Answer

$$8 	imes 7 	imes 6 = 336$$

Question 6

How many different 7-letter arrangements can be made from the letters in the word 'SUCCESS'?

Accepted Answer

$$\dfrac{7!}{3!\,2!\,2!} = 420$$

Question 7

A license plate consists of 2 letters (A-Z) followed by 3 digits (0-9). How many unique license plates are possible?

Accepted Answer

$$26^2$$ 	imes $$10^3 = 676$$,000$$

Question 8

A hospital needs to assign 4 nurses to 4 different shifts. If there are 7 nurses available, how many ways can the shifts be assigned?

Accepted Answer

$$P(7,4) = \dfrac{7!}{(7-4)!} = 840$$

Question 9

How many different ways can the letters in the word 'CALCULUS' be arranged?

Accepted Answer

$$\dfrac{8!}{2!\,2!\,2!} = 5040$$

Question 10

A medical research team needs to select 5 patients from a group of 20 for a clinical trial. In how many ways can the patients be selected if the order does NOT matter?

Accepted Answer

$$\dfrac{20!}{5!\,15!} = 15,504$$

Question 11

A password must be 4 characters long, using only the letters A, B, C, and D, with no letter repeated. How many different passwords are possible?

Accepted Answer

$$4! = 24$$

Question 12

A chemist has 10 different chemicals and wants to select 3 to mix together. In how many ways can the chemist choose the chemicals if the order does NOT matter?

Accepted Answer

$$\dfrac{10!}{3!\,7!} = 120$$

Question 13

How many different 8-digit codes can be made from 5 zeros and 3 ones?

Accepted Answer

$$\dfrac{8!}{5!\,3!} = 56$$

Question 14

A photographer wants to arrange 5 family members in a row for a photo. How many different arrangements are possible?

Accepted Answer

$$5! = 120$$

Question 15

A club has 13 members. How many ways can they elect a president, vice president, and treasurer (all different people)?

Accepted Answer

$$13 	imes 12 	imes 11 = 1,716$$

Question 16

How many ways are there to arrange the letters in the word 'BANANA'?

Accepted Answer

$$\dfrac{6!}{3!\,2!} = 60$$

Question 17

A team of 4 people is to be formed from a group of 8. How many different teams are possible?

Accepted Answer

$$\dfrac{8!}{4!\,4!} = 70$$

Question 18

A student is creating a 3-digit code using the numbers 1-9, but no digit can be repeated. How many different codes are possible?

Accepted Answer

$$9 	imes 8 	imes 7 = 504$$

Question 19

How many ways can you arrange 12 books on a shelf if 4 are blue, 3 are yellow, and 5 are white (books of the same color are identical)?

Accepted Answer

$$\dfrac{12!}{4!\,3!\,5!} = 27,720$$

Question 20

A teacher is choosing a line leader and a door holder from a class of 25 students. How many ways are there to choose these positions?

Accepted Answer

$$25 	imes 24 = 600$$

Counting Principles, Permutations, and Combinations in College Algebra

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