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

If $C=0$, every bin contains at most one ball, so the total number of balls is at most $m$. Since $n=dm>m$, the pigeonhole principle makes that event impossible:As $n\to \infty$ along these values, $m\to \infty$ and the mean occupancy $n/m$ is $d$. Apply part (b):Both powers tend to $e^{-d}$. HenceThe Poisson limit for occupancy fractions interprets the answer as the probability that a Poisson distribution variable with mean $d$ is at least two.