Expected Value of Max of Correlated Binomials

Hello all! I was wondering if there is a function to compute the expected value of N random variables by specifying their marginal distribution (in my case, Binomial(n,p), with n>1) and the matrix of correlations.

What I'm doing now is computing this number by simulating samples and taking averages. I'm looking for a more sophisticated way of doing this.

Thanks!