 Fourth centered moment of a binomial distribution

For $W\sim \mathrm{Bin}(n,p)$, expand the fourth power of the sum of centered independent Bernoulli variables. Terms with an index occurring once vanish; single-index fourth moments contribute $nq(1-3q)$ and paired second moments contribute $3n(n-1)q^2$. Their sum is the displayed formula. Scaling by $n^4$ yields a uniform $O(n^{-2})$ fourth moment for $W/n-p$.