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Video: Applications of the Counting Principle (Product Rule) and Combinations

Kathryn Kingham

Determine the number of ways a committee of 7 people, consisting of 5 boys and 2 girls, can be formed from a group of 9 boys and 4 girls.

02:57

Video Transcript

Determine the number of ways a committee of seven people, consisting of five boys and two girls, can be formed from a group of nine boys and four girls.

This is a combination problem. In combination problems, we say that order is not important. We say that because choosing Jake and Tim is the same thing as choosing Tim and Jake. The order here doesn’t matter.

Let’s start with the boys. How many ways can we choose five boys out of nine? We’re making five choices here. So we’ll start with nine, then we’ve eight choices, seven choices, six choices, and then five choices. But because the order doesn’t matter, we need to divide here. We’re gonna divide the top number, divide this numerator, by the number of ways we can arrange these five boys. We can arrange them in five times four times three times two times one ways.

Now I can do some simplification before we multiply and divide. The five divided by five cancels out. Three times two equals six. Six divided by six cancels out. We’ve eight divided by four, which can be reduced to two. We’re left with nine times two times seven. There are one hundred and twenty-six ways that we could choose our boys.

For our girls, how many ways can we choose two girls out of four? At first, we can choose from all four of them, and then we have three choices. Now we need to know how can we arrange those two girls which we have, two times one. Two times one ways to arrange them.

I can do some simplification here. Four over two can be reduced to two. Two times three is what we’re left with. Six ways to choose two girls out of four.

But our question is asking us how can we determine the seven people committee. To do that, we take our one hundred and twenty-six options for choosing a boy and our six options for choosing the girls and multiply them together.

There are seven hundred and fifty-six ways to choose a committee of seven people, if you’re choosing from nine boys and four girls.