Problem: A fair six-sided die is rolled 30 independent times, let $X$ be the number of ones and $Y$ the number of twos. Find the marginal PMF $P_Y(y)$ of $Y$.
Let $X=n, Y=m$, then the joint PMF is $$P_{X,Y}(x,y)=\binom{N}{n} \binom{N-n}{m} \left(\frac{1}{6}\right)^{n+m} \left(\frac{2}{3}\right)^{N-n-m}$$
Question: I know $P_Y(y)$ is of the form $\sum P_{X,Y}(x,y)$, but I am not sure how to represent $x$ and do the algorithm here..