Combination Subtraction (new)

There are $\binom{n+3}{3}$ ways of distributing $n$ identical objects among four people, and $\binom{n+2}{2}$ ways of distributing $n$ identical objects among three people. Thus we want $\begin{aligned} \binom{n+3}{3} - \binom{n+2}{2} & = \; 165 \\ \tfrac16(n+3)(n+2)(n+1) - \tfrac12(n+2)(n+1) & = \; 165 \\ \tfrac16n(n+1)(n+2) & = \; 165 \end{aligned}$ and hence $n = \boxed{9}$ .