Q: A teacher wants to select 4 students from a class of 25 to lead a discussion. In how many ways can this be done if the order in which they are chosen matters?

$$25 \times 24 \times 23 \times 22$$

Q: You have 8 shirts and want to choose 5 to wear over 5 days, wearing a different shirt each day. How many different ways can you do this?

$$8 \times 7 \times 6 \times 5 \times 4$$

Q: Emily has 15 shirts and space to hang 6. How many ways can she hang shirts in that section?

$$15 \times 14 \times 13 \times 12 \times 11 \times 10$$

Q: A student must answer 10 fill-in-the-blank questions, but can only use a word bank of 14 words, using each word once. How many possible ways can the student answer all 10 questions?

$$14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5$$

Question 1

Which of the following scenarios best illustrates a permutation rather than a combination?

Accepted Answer

Assigning president, vice president, and treasurer roles to 3 students from a group of 13.

Question 2

What is the formula for the number of permutations of $$n$$ objects taken $$r at a $$time?

Accepted Answer

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

Question 3

A teacher wants to select 4 students from a class of 25 to lead a discussion. In how many ways can this be done if the order in which they are chosen matters?

Accepted Answer

$$25 	imes 24 	imes 23 	imes 22$$

Question 4

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

Accepted Answer

$$\frac{6!}{3!2!}$$

Question 5

You have 8 shirts and want to choose 5 to wear over 5 days, wearing a different shirt each day. How many different ways can you do this?

Accepted Answer

$$8 	imes 7 	imes 6 	imes 5 	imes 4$$

Question 6

A student council has 13 members and needs to elect a president, vice president, and treasurer. How many ways can these positions be filled?

Accepted Answer

$$13 	imes 12 	imes 11$$

Question 7

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

Accepted Answer

$$\frac{8!}{5!3!}$$

Question 8

When arranging books by color, you have 12 books: 4 blue, 3 yellow, and 5 white. How many different ways can you arrange them?

Accepted Answer

$$\frac{12!}{4!3!5!}$$

Question 9

Which of the following is the correct formula for the number of combinations of $$n$$ objects taken $$r at a $$time?

Accepted Answer

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

Question 10

A lottery game requires you to choose 5 numbers from 1 to 40. What is the total number of possible tickets?

Accepted Answer

$$\frac{40!}{5!35!}$$

Question 11

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

Accepted Answer

Question 12

Which of the following statements is TRUE regarding permutations and combinations?

Accepted Answer

Permutations are used when order matters.

Question 13

Emily has 15 shirts and space to hang 6. How many ways can she hang shirts in that section?

Accepted Answer

$$15 	imes 14 	imes 13 	imes 12 	imes 11 	imes 10$$

Question 14

A teacher wants to select 4 students from a class of 28 to lead a discussion. If the order does not matter, how many ways can this be done?

Accepted Answer

$$\frac{28!}{4!24!}$$

Question 15

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

Accepted Answer

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

Question 16

A student must answer 10 fill-in-the-blank questions, but can only use a word bank of 14 words, using each word once. How many possible ways can the student answer all 10 questions?

Accepted Answer

$$14 	imes 13 	imes 12 	imes 11 	imes 10 	imes 9 	imes 8 	imes 7 	imes 6 	imes 5$$

Question 17

In a lottery game, you must choose 5 numbers from 1 to 40. What is the probability of winning if only one ticket matches the winning numbers?

Accepted Answer

$$\frac{1}{\frac{40!}{5!35!}}$$

Question 18

A hotel has 32 floors. You need to pick 2 floors to blend into a milkshake. How many possible ways can you select two floors?

Accepted Answer

$$\frac{32!}{2!30!}$$

Counting Principles: Permutations, Combinations, and Applications in Statistics

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