In how many different ways can 12 people be picked from 15?
In this question, we’re looking to choose 12 people from a total of 15. There’s no indication that the order in which we choose these 12 people matters. In mathematics, we call this a combination. It’s a way to calculate the total outcomes of an event where the order of the outcomes does not matter. And if we’re looking to find the number of ways of choosing our items from a total of 𝑛 items, we use 𝑛 choose 𝑟, where 𝑛 choose 𝑟 is 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. And 𝑛 factorial is 𝑛 times 𝑛 minus one times 𝑛 minus two times 𝑛 minus three, and so on.
Now, we’re looking in this question to choose 12 people from a total of 15. So, in our formula, we’re going to let 𝑟 be equal to 12 and 𝑛 be equal to 15. And so, the calculation we need to perform is 15 choose 12. That’s 15 factorial over 12 factorial times 15 minus 12 factorial. But of course, 15 minus 12 is three. So, we have 15 factorial over 12 factorial times three factorial.
Now, generally, when we’re calculating combinations, we want to try and avoid evaluating our factorials fully. So, we wouldn’t really want to work out 15 times 14 times 13 times 12, and so on. Instead, we spot that 15 factorial can be written as 15 times 14 times 13 times 12 factorial. And then, we see that we can divide both our numerator and denominator by 12 factorial. We need to look for some further common factors. Well, we can divide both 14 and two by two and 15 and three by three. And so, 15 choose 12 simplifies to five times seven times 13 divided by one or just five times seven times 13. This is 455.
And so, by calculating 15 choose 12, we’ve worked out the number of ways that 12 people can be picked from 15. It’s 455.