We can describe the procedure of distributing the presents as follows:
- First, we select $n_{1}$ presents and give them to the first child. This can be done in $\binom{n}{n_{1}}$ ways.
- Then we select $n_{2}$ presents from the remaining $n − n_{1}$ and give them to the second child, etc$\ldots$
Complete this argument and show that it leads to the same result as the previous one. I think that this will yield a sum of binomials $\binom{n}{n_{1}} + \binom{n - n_{1}}{n_{2}} + \binom{n - n_{1} - n_{2}}{n_{3}}$ and so on. The result of the previous example is $$\frac{n!}{n_{1}!\,n_{2}!\ldots n_{k}!} $$ However, I'm not sure where to go after finding the sum of the binomials. Thank you so much !.