Suppose we throw n balls into m bins, such that the ball locations are uniform and mutually independent. For (a) and (b) use the fact that the bin choices for different balls are mutually independent.
a) The expected number of balls in a particular bin is n/m
b) The variance of the number of balls in a particular bin is n(1-1/m)
c) The probability that a particular bin is empty is (1-1/m)ⁿ
d) The probability that all bins are occupied is (1/m)ⁿ