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

Express the three counts as sums of indicator variables for $B_i=0$, $B_i=1$ and $B_i\ge 2$. The linearity of expectation applies even though the bin counts are dependent. The binomial distribution givesThusThe relation $E+S+C=m$ checks the sum of these expectations. For $m=1$, use the direct deterministic occupancies, or the usual binomial convention $0^0=1$ when $n=1$.