How many Combinations there are to choose 5 Men and 5 Women and make them a pair?

Question

There are 22 students - 10 Men and 12 Women.

I have to choose randomaly 5 Men and 5 Women.

How many Combinations there are to choose 5 Men and 5 Women and make them a pair ?

I know i have to start with $\binom{10}{5}\binom{12}{5}$, But I dont know how to continue.

That's a reasonable start. It is maybe easier to choose the $5$ women, line them up in order of student number, and have them choose their partners. — André Nicolas, Mar 15 '16 at 17:49

score 3 · Answer 1 · answered Mar 15 '16 at 17:50

I'll take "make them a pair" to mean "pair them off," or "pair each man with one woman."

If so, you're almost done. You've chosen five men and five women, and counted the possibilities correctly.

Now, to pairing them off. Number the men $1$ through $5$. Man $1$ can be paired off five ways (to any of the women). Man $2$ can then be paired off four ways (to any of the remaining women).

Can you take it from here?

How many Combinations there are to choose 5 Men and 5 Women and make them a pair?

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