Let \((X_1, \ldots, X_k)\sim\text{Multinomial}(n, p_1, \ldots, p_k)\). I'll show how to calculate \(Cov(X_i, X_j), i\ne j\). The main purpose of doing this is to illustrate the little trick of reducing complexity of problems by decomposing a complicated random variable into simple ones.

Let \((Y_ …