I'm reviewing for my stat midterm and I happen to block at this question. I tried to choose out of the women 5 and out of the men 5 multiply both and divide by the total (17 choose 10 since 5 men and 5 women).
At a party, a person is introduced to 10 women and 7 men. What is the probability that she identifies all the couples correctly if she knows that there are 5 couples? [Assume a couple is a man and a woman].
There are $\binom{10}{5}\times \binom{7}{5}\times 5!$ different number of ways in which you may select five women, select five men, and choose a way in which to pair them together. Only one of these ways will be correct.
To count this, again, choose which five women we suspect are in one of the couples relationships. Then choose which five men we suspect are in the couples relationships. Then, among our selected women from youngest to oldest choose we we think is the male partnered with the woman.
The answer then:
$$\dfrac{1}{\binom{10}{5}\times\binom{7}{5}\times 5!}$$