PUMaC 2016 Team Problem 8

Alice has 100 balls and 10 buckets. She takes each ball and puts it in a bucket that she chooses at random. After she is done, let b_{i} be the number of balls in the i th bucket, for 1 \leq i \leq 10. Compute the expected value of \sum_{i=1}^{10} b_{i}^{2}.