An urn has $N_1$ red, $N_2$ green and $N_3$ yellow balls. $$N = N_1 + N_2 + N_3,$$ which denotes the total number of balls in the urn. $$p_1 = \frac{N_1}{N},\ p_2 = \frac{N_2}{N},\ p_3 = \frac{N_3}{N},$$ which denotes the proportions of red, green and yellow balls in the urn. A simple random sample of $n$ balls is taken from the urn, where $n < N_i$, $i =1,2,3$.
I need to obtain an approximation to this probability when $n$ is much smaller than $N_1, N_2, N_3$.
I know that in a situation that only has $N_1$ and $N_2$, the approximation to the probability when $n$ is much smaller than $N_1$ and $N_2$ is:$$\frac{n!}{x!(n - x)!} × p^x × (1 - p)^{n-x}.$$
However, for the life of me, I cannot figure out how to do with $N_1, N_2, N_3$. Any help would be amazing.
\begin{eqnarray*} (p_1+p_2+p_3)^{N}= \sum_{N_1+N_2+N_3=N} \frac{N!}{N_1 ! N_2 ! N_3 !} p_1^{N_1} p_2^{N_2} p_3^{N_3}. \end{eqnarray*}