What is the distribution of the number of heterozygotes in finite populations?
It is a multinomial (trinomial) distribution with $N$ trials, $p_{AA}=p^2$, $p_{Aa}=2p(1-p)$, and $p_{aa}=(1-p)^2$. The probability to get exactly $n_{AA}$ AA genotypes, $n_{Aa}$ Aa genotypes and $n_{aa}=N-n_{AA}-n_{Aa}$ aa genotypes is 

$$P\left(N_{AA} = n_{AA}, N_{Aa} = n_{Aa}\right) = \frac{N!}{n_{AA}!n_{Aa}!n_{aa}!}p^{2n_{AA}}\left(2p(1-p)\right)^{n_{Aa}}(1-p)^{2n_{aa}}.$$