How many teams of four can be selected from a group of 20 people?
Before we start calculating anything, let’s answer some questions. First, we want to know does order matter? Does the order in which we choose the team members matter? Yes or no? If order does matter, we’re dealing with the permutation. If order does not matter, we’re dealing with the combination. The order of choosing this team does not matter. So we’ll follow the combination route.
After that, we’ll need to consider if repetitions are allowed. In the context of this question, “are repetitions allowed?” would be is one person allowed to take up more than one space on the team? If these are our four slots and Bob is selected for slot A, could Bob also be selected for slot B? No, he can only take one spot on the team.
This tells us that repetition is not allowed in this scenario, which means we need to use the combination formula, 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial, where 𝑛 equals the number of options and 𝑟 equals the number of slots. The number of options in our case is 20. We’ve 20 people to choose from. 𝑟, the number of slots, is four. There are four spaces on the team.
Our formula when we plug everything in would look like this: 20 factorial over four factorial times 20 minus four factorial. 20 minus four equals 16. We can break up our 20 factorial into two pieces. We can say 20 times 19 times 18 times 17 times 16 factorial. Why would we do that? When we break it up like that, we have 16 factorial in the numerator and 16 factorial in the denominator. And they cancel out.
If we expand our four factorial in the denominator, we’ll now have four times three times two times one. And we can do some simplifying. 20 divided by four equals five. 18 divided by three equals six. And six divided by two equals three. We have five times 19 times three times 17 options, which equals 4845.
In our context, it means that there are 4845 ways to choose a team of four out of 20 people.