Snowball Question FFT
Here is the idea of the proof. Let $a_i,b_i$ be the distance thrown by the $i$th male/female. Using FFT, we calculate
$$ \left(\sum_i x^{a_i}\right) \left(\sum_j x^{b_j}\right) = \sum_{i,j} x^{a_i+b_j}. $$
Each pair $i,j$ satisfying $a_i + b_j = k$ contributes one term $x^k$ to the polynomial on the right. Hence the coefficient of $x^k$ is the number of pairs $i,j$ such that $a_i + b_j = k$.

If things still aren't clear, I suggest you try a few examples.