We went over counting techniques previously. Now, let’s go over some more involved problems of that nature.

• Eric manages 5 employees. Each of them works on one of the days Monday through Friday. They each work only one day. How many ways are there to choose an ordering of employees for a workweek?

Let’s go through this day by day. On the first day, there are 5 options for an employee. On the second day, there are 4 options. This pattern continues until the last day when there is only one option.

So, the total number of options is 5 x 4 x 3 x 2 x 1 = 120. This is equivalent to 5 factorial, which is also written as 5!.

A factorial involves multiplying the number and all integers smaller than it until 1. Factorials are extremely useful and prevalent in combinatorics.

• How many ways are there to arrange the letters of MOEMS?

Let’s use a similar approach to the previous problem. There are 5 options for the first spot, 4 for the second spot, etc. The total number of arrangements is 5!, which is 120.

Wait! Let’s check if we are overcounting. Overcounting is when we count the same arrangement as two different arrangements.

In this problem, we are overcounting. We are counting the two Ms as unique when they are the same letter. For example, the arrangement of MOEMS is still the same if we switch the two Ms.

Thus, to get the actual answer, we need to divide 120 by the number of ways to switch the two Ms, which is 2. So, the final answer is 60.

• How many ways are there to arrange the letters of Mississippi?

This problem is similar to the previous problem, but there are more letters and repeats. Let’s work through it.

There are 11 letters, so there are initially 11! arrangements without accounting for overcounting. There are 4 Is, so there are 4! ways to arrange them. There are 4 Ss and 4! ways to arrange them. Finally, there are 2 Ps and 2! Ways to arrange them.

Thus, the final answer is 11! / (4! X 4! X 2!). This evaluates to 34650. You will likely never have to do a computation with super-large numbers like this on an actual test.