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I have a problem, 50 people consisting of 25 boys and 25 girls are going to send 20 people to a conference. How many ways can you select 20 people so that there are 10 girls and 10 boys?

This is where I get confused with these types of problems. I believe it would be a combination problem cause order doesn't matter in this case. (could be wrong about that) Also, would I set it up as 50C10 multiplied by 50C10? By 50C10 I mean the formula for combinations of nCk, where I am choosing 10 from a set of 50. Maybe I am just overthinking the problem cause I am new to the concept but some help would be appreciated. Am I correct, or way off?

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2 Answers

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You are right that it's a combination since order doesn't matter. But remember you're choosing $10$ girls/boys from $25$ and not from $50$. So it's ${25\choose 10}{25\choose 10}.$

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First choose the $10$ boys out of the $25$ boys to be in the conference. Then choose the $10$ girls out of the $25$ girls to be in the conference. These are independent choices. Hence there are $$ \binom{25}{10}\binom{25}{10} $$ ways.

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