Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 10F / c / Solution

There are more bins than balls, so the pigeonhole principle forces at least one empty bin. In fact $E\ge m-n=(a-1)n$. ConsequentlyFor the expected fraction, substitute $m=an$ into part (b). The elementary exponential limit, obtained by taking logarithms, givesThis is the empty-bin case of the Poisson limit for occupancy fractions, with limiting mean occupancy $1/a$.