Q: A radio station call sign must start with either 'K' or 'W', followed by exactly three letters (A-Z), with repetition allowed. How many possible call signs are there?

$$2 \times 26 \times 26 \times 26 = 35,152$$

Q: A password consists of 2 letters (A-Z) followed by 3 digits (0-9), with repetition allowed. How many possible passwords are there?

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

Question 1

A university cafeteria offers 4 vegan entrees and 6 non-vegan entrees. If a student chooses exactly one entree, how many possible choices are there?

Accepted Answer

$$4 + 6 = 10$$

Question 2

A security system uses a 5-digit code, where each digit can be any number from 0 to 9, and digits can repeat. How many possible codes are there?

Accepted Answer

$$10^5 = 100$$,000$$

Question 3

A company must select a 4-person team from 15 employees. The order in which they are selected does not matter. How many different teams are possible?

Accepted Answer

$$15C4 = \frac{15!}{(15-4)!\ 4!}$$

Question 4

How many distinct ways can the letters in the word 'BALLOON' be arranged?

Accepted Answer

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

Question 5

A radio station call sign must start with either 'K' or 'W', followed by exactly three letters (A-Z), with repetition allowed. How many possible call signs are there?

Accepted Answer

$$2 	imes 26 	imes 26 	imes 26 = 35,152$$

Question 6

A basketball coach wants to determine the shooting order for 5 players. How many possible orders are there?

Accepted Answer

$$5! = 120$$

Question 7

A technician needs to test all possible pairs of wires from a cable containing 10 color-coded wires. How many unique pairs can be tested?

Accepted Answer

$$10C2 = \frac{10!}{8!\ 2!} = 45$$

Question 8

A student is creating a password using 3 letters (A-Z), where letters can repeat. How many possible passwords are there?

Accepted Answer

$$26^3 = 17$$,576$$

Question 9

From a group of 8 runners, medals are awarded for 1st, 2nd, and 3rd place. How many possible ways can the medals be distributed?

Accepted Answer

$$8P3 = \frac{8!}{5!} = 336$$

Question 10

A word has 9 letters, with 3 of them being 'A', 2 being 'B', and the rest unique. How many distinct arrangements are possible?

Accepted Answer

$$\frac{9!}{3!\ 2!}$$

Question 11

A moving truck must deliver to 7 different sites. If the delivery order is randomly selected, what is the probability that it happens to be the shortest possible route?

Accepted Answer

$$\frac{1}{7!}$$

Question 12

A committee of 5 is to be formed from 20 people. What is the probability that a randomly selected committee consists of the 5 oldest people?

Accepted Answer

$$\frac{1}{20C5}$$

Question 13

A password consists of 2 letters (A-Z) followed by 3 digits (0-9), with repetition allowed. How many possible passwords are there?

Accepted Answer

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

Question 14

Which formula would you use to count the number of ways to select 4 books from a shelf of 12, where the order does not matter?

Accepted Answer

$$12C4 = \frac{12!}{8!\ 4!}$$

Question 15

A word has 8 letters, with 2 'E's and 3 'T's. How many distinct arrangements are possible?

Accepted Answer

$$\frac{8!}{2!\ 3!}$$

Question 16

A student must choose one science course from 3 physics options or 4 chemistry options. How many possible choices does the student have?

Accepted Answer

$$3 + 4 = 7$$

Question 17

A lottery requires picking 6 numbers from 49, where order does not matter. How many possible tickets are there?

Accepted Answer

$$49C6 = \frac{49!}{43!\ 6!}$$

Question 18

A word has 10 letters, with 4 'A's and 2 'B's. How many distinct arrangements are possible?

Accepted Answer

$$\frac{10!}{4!\ 2!}$$

Question 19

A student is picking an outfit: 3 shirts, 2 pants, and 4 pairs of shoes. How many possible outfits can be made?

Accepted Answer

$$3 	imes 2 	imes 4 = 24$$

Counting Principles in Probability: Sums, Products, Permutations, and Combinations

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