### Video Transcript

A school holds a ceremony to honor members of the school administration, which contains 25 teachers and 15 executives. The ceremony will honor 10 members, and the school must choose at least five teachers and at least three executives. Given that the teachers will stand on the left of the stage and the executives will stand on the right of the stage, what is the total number of possible onstage configurations for the members of the school administration? Express your answer in terms of combinations or permutations.

Let’s begin by thinking about what this could look like. We’re going to choose at least five teachers from a total of 25 and at least three executives from a total of 15. Our aim in doing so is to honor exactly 10 members. So how many teachers and how many executives could we have? Well, we can have no fewer than five teachers. So we could have exactly five teachers, which would mean we would also have exactly five executives. We could have six teachers and four executives. And the final option is to have seven teachers and three executives. We know there are no other options because we must choose at least three executives.

We’re also told that, as in the diagram, the teachers will stand on the left of the stage and the executives will stand on the right. This minimizes the total number of ways of ordering these groups of people. In fact, we’ve represented all three possible ways of doing so. It does not matter the order in which the teachers stand, nor does it matter the order in which the executives stand. And so, that should remind us whether we need to use combinations or permutations here. The difference between calculating combinations and permutations is whether order matters. With a combination, the order here in which the people stand matters. With permutations, the order in which our people stand doesn’t matter. So we’re going to be using permutations.

And if we want to calculate the number of ways of choosing 𝑟 items from a total of 𝑛 when order doesn’t matter, we represent that as 𝑛P𝑟 as shown. So if we look at the number of ways of choosing our five teachers, we know we’re choosing from a total of 25. Since order doesn’t matter then, the number of ways of choosing these five teachers is 25P five. In a similar way, we’re choosing five executives from a total of 15, and order doesn’t matter, so that’s 15P five. Then in our next option, it’s 25P six to choose six teachers and 15P four to choose four executives. Finally, choosing seven teachers from a total of 25 is 25P seven. And to choose our three executives is 15P three.

But how do we combine these? Well, here’s where we remind ourselves what the product rule for counting tells us. This tells us that the total number of outcomes for two or more events is found by multiplying the number of outcomes of each event together. So the number of ways of choosing five teachers and five executives is 25P five times 15P five. Then the number of ways of choosing six teachers and four executives is 25P six times 15P four. And finally, we multiply 25P seven by 15P three to find the total number of ways of choosing seven teachers and three executives. We’re still not done, of course. We still need to combine each of these three options.

This is where we remind ourselves what we mean when we talk about the addition rule. Sometimes called the basic counting principle, it tells us that if we have 𝑎 number of ways of doing something and 𝑏 number of ways of doing another thing and we can’t do both at the same time, then there are 𝑎 plus 𝑏 ways to choose one of the actions. In other words, to find the total number of ways of choosing five teachers and five executives or six teachers and four executives or seven teachers and three executives, we add each of these together. And that tells us the total number of onstage configurations for members of this school administration. And that’s an answer given in permutations. It’s 25P five times 15P five plus 25P six times 15P four plus 25P seven times 15P three.