Yes he did, in two parts: easy non-brute-force ways exist (as in the Monte-Carlo integration in R, he argues why it isn't brute force) and also a non-easy analytical solution exists (when at least one set of parameters are integer, in the linked cross validated post).

He also explains that easy analytical solutions often don't exists to seemingly easy questions, such as this one. That's why Bayesian statistics needs Monte-Carlo.