If $(X_1,\cdots, X_n)$ is a vector with multinomial distribution, proof that $\text{Cov}(X_i,X_j)=-rp_ip_j$, $i\neq j$ where $r$ is the number of trials of the experiment, $p_i$ is the probability of success for the variable $X_i$. $$fdp=f(x_1,...x_n)={r!\over{x_1!x_2!\cdots x_n!}}p_1^{x_1}\cdots p_n^{x_n} $$ if $ x_1+x_2+\cdots +x_n=r$
I'm trying to use the property: $\text{Cov}(X_i,X_j)=E[X_iX_j]-E[X_i]E[X_j]$ and find that $E[X_i]=rp_i$, but I don´t know the efficient way to calculate $E[X_iX_j].$