Permutation & Combination: Problems & Practice

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will practice solving various permutation and combination problems using permutation and combination formulas. We can continue our practice when we take a quiz at the end of the lesson.

Permutations and Combinations

Both permutations and combinations are groups or arrangements of objects. However, there is one big difference between them. When dealing with combinations, the order of the objects is insignificant, whereas in permutations the order of the objects makes a difference.

For example, assume you have 10 coins in your pocket and you take 5 out, a dime, 2 quarters, a nickel and a penny. If I said you grabbed those same 5 coins, but I said you grabbed 2 quarters, a nickel, a penny, and a dime, it is still the same group of coins. That is, the order I name them in is insignificant. Therefore, the coins are a combination of 5 of 10 coins.

Now consider the scenario where we are talking about 5 finishers in a race, runners A, B, C, D, and E. If I tell you they crossed the line in the order A, B, C, D, E, this would be different than if I told you they crossed the line in the order C, B, A, E, D. Thus, the order makes a difference, so the order in which the 5 runners finish is a permutation of the 5 runners.

Solving Permutation and Combination Problems

There are two questions you have to answer before solving a permutation/combination problem.

1.) Are we dealing with permutations or combinations? In other words, does order matter?

2.) Is repetition allowed? In other words, can we name an object more than once in our permutation or combination?

Once we have answered these questions, we use the appropriate formula to solve the problem. Those formulas are shown below.

permutation formulas

combination formulas

Practice Problems

Let's look at some examples to get comfortable solving these types of problems.

1.) How many distinct ways can the letters of the word FRIEND be arranged?

Solution: We will start off by determining if we're dealing with permutations or combinations. To do this, consider the arrangement FRIEND and the arrangement DIREFN. We see that these represent two different arrangements, so the order of the letters make a difference. Thus, we're dealing with permutations. Now we must consider if repetition is allowed. When we create an arrangement of the letters, we put one letter in the first spot, then the second spot, and so on until all 6 spots are filled. Once we put a letter in a spot, we can't put it in another spot, because it has already been used. Therefore, repetition is not allowed. We know that we are dealing with a permutation of 6 objects (letters) where repetition is not allowed. Therefore, we use formula n! We plug 6 in for n in the formula to get the following.

6! = 6 * 5 * 4 * 3 * 2 * 1 = 720

We see that there are 720 ways to arrange the letters in the word FRIEND.

2.) Assume you must select 3 people from a group of 20 people. How many ways are there to do this?

Solution: If we select 3 people, say George, Frank, and Jessica, it is still the same group if we said we selected Frank, Jessica, and George. Therefore, order is insignificant, so we are dealing with combinations. Once we have selected a person, we can't select them again since they are already in the group. Therefore, repetition is not allowed. Based on this information, we use the formula nCr = n! / r!(n - r)!, where we plug in 3 for r and 20 for n. Doing so gives the following.

20! / 3!(20 - 3)! = 20! / 3!17! = (20*19*18) / (3*2*1) = 1,140

Therefore, there are 1,140 ways to choose 3 people from a group of 20.

3.) How many 3-digit numbers can be formed from the digits 3, 7, 0, 2, and 9?

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