There are n balls $b_1...b_n$ and $n$ boxes. Each ball is placed in a box chosen independently and uniformly at random.We say that $(b_i,b_j)$ is a colliding pair if $i<j$ and $b_i$ and $b_j$ are placed in the same box.Find the expected number of colliding pair.
The p.d.f of choosing a box should be $\frac{1}{n}$, now if n balls need to be distributed into n box shouldn't we use $n+n-1\choose n $, and then find expected value. But the answer given is $\frac{n-1}{2}$, how should I do this?