I came across an exercise in which we are asked to prove the identity:
$${2n\choose n}=\sum_{k=0}^n{n\choose k}^2$$
The exercise gives the hint:
$$\left(1+x\right)^{2n}=\left[(1+x)^n\right]^2$$
It's not too difficult to use the hint to prove the identity (the expressions in the identity are the coefficients of $x^n$ in the respective expansions of the expressions in the hint, which of course must be the same number), but I was wondering whether there is a neater equivalent-counting interpretation...
It's clear that ${2n \choose n}$ is the number of ways in which we can choose half the elements in a set (where this is possible): how can we interpret $\sum_{k=0}^n{n\choose k}^2$ equivalently?