The following problems will test your knowledge of permutations and combinations.

**Problem 1: A club of eight people are selecting a board of two people: a president and a vice president. How many ways are there to choose the board from the eight people?**

**Step 1: Identify whether we are finding permutations or combinations. **

Since each position on the board is unique, we are finding permutations.

**Step 2: Find the total number of permutations.**

**Using the visual approach:**

There are 8 people who could be chosen as president. Then once the president has been chosen, there are 7 remaining people who could be chosen as vice president. To find the total number of permutations, we simply multiply these numbers together. Thus, there are 56 different ways to choose a board of two people from the eight people.

**Using the formula:**

_{n}P_{r} = n! / (n-r)!

In this case, n = 8 and r = 2. Plugging these into the formula we get:

_{8}P_{2} = 8! / (8-2)! = 8! / 6! = 40,320 / 720 = **56**

**Problem 2: You won a trip to Japan and you’re allowed to bring three friends with you. Unfortunately, six of your friends want to come along. How many different groups of friends could you take with you?**

**Step 1: Identify whether we are finding permutations or combinations. **

Since the order of friends you bring is not important (i.e. the group of “Duane, Steve, and Brad” is the same as the group of “Brad, Steven, and Duane”), we are finding combinations.

**Step 2: Find the total number of combinations.**

In this case, there are six total friends (n = 6) and you want to select a group of three (r = 3). Using the formula:

Total combinations = _{6}C_{3 } = 6! / 3!(6-3)! = 5! / 3!(3!) = (6*5*4*3*2*1) / (3*2*1)(3*2*1) = 720 / 36 = **20**

There are 20 different groups of friends you could take with you to Japan.

**Problem 3: You have a bookshelf in your room that can hold three books. You have seven books sitting on your bed to choose from that you can place on the bookshelf. How many different ways can you select a group of three books?**

**Step 1: Identify whether we are finding permutations or combinations. **

Since the order of books is not important (i.e. the group of “book 1, book 2, book 3” is the same as the group of “book 3, book 2, book 1”), we are finding combinations.

**Step 2: Find the total number of combinations.**

In this case, there are seven total books (n = 7) and you want to select a group of three (r = 3). Using the formula:

Total combinations = _{7}C_{3 } = 7! / 3!(7-3)! = 7! / 3!(4!) = (7*6*5*4*3*2*1) / (3*2*1)(4*3*2*1) = 5,040 / 144 = **35**

There are 35 different ways you can select a group of three books.

**Problem 4: A football coach is recruiting three players for the following positions: quarterback, receiver, and running back. There are eight players trying out for these positions. How many ways are there to choose three players from this group of eight players?**

**Step 1: Identify whether we are finding permutations or combinations. **

Since each position on the team is unique, we are finding permutations.

**Step 2: Find the total number of permutations.**

**Using the visual approach:**

There are 8 people who could be chosen as quarterback. Then once the quarterback has been chosen, there are 7 remaining people who could be chosen as receiver. Then once a quarterback and receiver have both been chosen, there are 6 remaining people who could be chosen as running back. To find the total number of permutations, we simply multiply these numbers together. Thus, there are 336 different ways to choose three players from this group of eight.

**Using the formula:**

_{n}P_{r} = n! / (n-r)!

In this case, n = 8 and r = 2. Plugging these into the formula we get:

_{8}P_{3} = 8! / (8-3)! = 8! / 5! = 40,320 / 120 = **336**

**Problem 5: A researcher wants to select two students from a classroom of seven students to take a survey. Both students will take the exact same survey. How many ways are there to choose two students from the classroom of seven students?**

**Step 1: Identify whether we are finding permutations or combinations. **

Since the students are taking the same survey and the order of the students does not matter (i.e. the group of “Jessica and Marie” is the same as the group of “Marie and Jessica”), we are finding combinations.

**Step 2: Find the total number of combinations.**

In this case, there are seven total students (n = 7) and we want to select a group of two (r = 2). Using the formula:

Total combinations = _{7}C_{3 } = 7! / 2!(7-2)! = 7! / 2!(5!) = (7*6*5*4*3*2*1) / (2*1)(5*4*3*2*1) = 5,040 / 240 = **21**

There are 21 different ways to select a group of two students.

**Problem 6: A group of five people are trying out for a band that only has two open roles: lead singer and backup singer . How many different ways are there to choose two roles from these five people?**

**Step 1: Identify whether we are finding permutations or combinations. **

Since each role in the band is unique, we are finding permutations.

**Step 2: Find the total number of permutations.**

**Using the visual approach:**

There are 5 people who could be chosen as lead singer. Then once the lead singer been chosen, there are 4 remaining people who could be chosen as backup singer. To find the total number of permutations, we simply multiply these numbers together. Thus, there are 20 different ways to choose two roles from this group of five.

**Using the formula:**

_{n}P_{r} = n! / (n-r)!

In this case, n = 8 and r = 2. Plugging these into the formula we get:

_{5}P_{2} = 5! / (5-2)! = 5! / 3! = (5*4*3*2*1) / (3*2*1) = **20**