I'm stuck on a question finding the maximum likelihood estimators of
$$p(x,y)=\dfrac{n!}{x!\cdot y!\cdot (n-x-y)!\cdot }{p^x\cdot q^y\cdot (1-p-q)^{n-x-y}}$$
I think it is a multinomial distribution but am struggling to get the likelihood function because of the number of variables. How would you get the likelihood function for this?
You should change the name of the pmf as it conflicts with the names of the parameters. In this case $$ f(x,y,p,q)=\frac{n!}{x!y!(n-x-y)!}p^yq^y(1-p-q)^{n-x-y}. $$ The Likelihood is the pmf viewed as a function of $p$ and $q$ for fixed $x$ and $y$. In other words $$ L_{x,y}(p,q)=\frac{n!}{x!y!(n-x-y)!}p^yq^y(1-p-q)^{n-x-y}\tag{1}. $$ If you wish to find the log-likelihood $\ell(p,q)$, simply take the logarithm of (1).