Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 12I / i / Solution

The columns of the parity-check matrix of the binary Hamming code are precisely the $n=2^d-1$ distinct nonzero vectors of ${\mathbb{F}}_2^d$. If $h_1\ne h_2$, then $h_1+h_2$ is nonzero and differs from both summands, so it occurs as a unique third column $h_3$.

A word supported on three positions is a codeword exactly when the corresponding three columns sum to zero. Every unordered pair of columns determines one such triple, and each triple is counted once by each of its three pairs. Therefore