Calculate Covariance Between Multinomial Categories by Hand

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 $(Y_{i1}, \ldots, Y_{ik})\overset{iid}{\sim}\text{Multinomial}(1, p_1, \ldots, p_k), i\ge1$, then

So,