Probability to form boy-girl pairs n pairs are formed from n girls and n boys randomly. What is the probability that each pair is formed by one boy and one girl?

Line up the girls, say by student number, and let them choose a partner one at a time. The probability the first girl chooses a boy is $\frac{n}{2n-1}$. Given that she chooses a boy, the probability the second girl chooses a boy is $\frac{n-1}{2n-3}$. 
Given that the first $2$ girls each choose a boy, the probability the third girl chooses a boy is $\frac{n-2}{2n-5}$. And so on. Now take the product.
Remark: The answer can be made more compact. At the bottom we have $(2n-1)(2n-3)(2n-5)\cdots (1)$. This is $\frac{(2n)!}{2^n n!}$, so an alternative form of the answer is 
$\frac{2^n (n!)^2}{(2n)!}$. 
