A homework question asks:
Let $(X, Y, Z)$ have a multinomial distribution with parameter $n = 3, \ p1 = \frac{1}{6}, \ p2 = \frac{1}{2}, \ p3 = \frac{1}{3}$.
Find cov$(X, Y)$.
Hint: first find the joint p.m.f. of $X$ and $Y$.
The given answer is: -.25
However I cannot find anything that teaches us on how to get the joint pmf of 2 variables when it is in a distribution with 3. If you could help me understand how to get the joint pmf of X and Y, the rest of the problem wouldn't be hard.
For any $k\geq 0$, $m\geq 0$, $k+m\leq 3$ $$\mathbb P(X=k, Y=m)=\mathbb P(X=k, Y=m, Z=3-k-m)=\dfrac{3!}{k!m!(3-k-m)!}\left(\frac{1}{6}\right)^{k}\left(\frac{1}{2}\right)^{m}\left(\frac{1}{3}\right)^{3-k-m}.$$