Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 16 / a / Solution

A uniformly shuffled deck makes all unordered five-card hands equally likely, giving $\left(\genfrac{}{}{0ex}{}{52}{5}\right)$ possible hands. Choose the rank appearing four times in $13$ ways; all four suits of that rank are then fixed. The remaining card can be any of the other $48$ cards. No hand is counted twice, because five cards cannot contain two different complete four-card ranks. This use of combinatorics gives the probabilityHere $\left(\genfrac{}{}{0ex}{}{52}{5}\right)=2598960$ is a binomial coefficient; the probability also simplifies to $1/4165\approx 0.000240096$.