We are putting balls into bins by the Uniform Distribution. Let $\displaystyle X$ be a Random Variable and means moment of collision when two balls fall into the same bin. So minimum $\displaystyle X$ is 2 (2 balls in the same bin) and maximum $\displaystyle X$ is $\displaystyle n+1$. What is probability $\displaystyle P(2 \le X \le n+1)$ and Expected Value for $\displaystyle k$ bins and $\displaystyle n$ balls?