(a) Let n be a nonnegative integer. Use the identity

$\displaystyle (1 + x)^n(1 + x)^n = (1 + x)$^$\displaystyle (2n)$

to show that$\displaystyle \sum_{n=0}^{n}\binom{n}{k}^2 =\binom{2n}{n}$ (1)

(b) Prove (1) again by counting, in two different ways, the number of ways

of choosing n people from a set of n girls and n boys.