9. A convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. For $V$ vertices, each vertex has $T$ triangular faces and $P$ pentagonal faces intersecting. Find the value of $P+T+V$.


9. Solution: Since $F=32$, according to Euler's formula, we have $32+V-E=2$, so $E=V+30$. Because $T+P$ faces intersect at each vertex, there are $T+P$ edges emanating from each vertex, thus $2E=V(T+P)$. Therefore, $V(T+P)=2(V+30)$, which simplifies to $V(T+P-2)=60$. Since each triangular face has 3 edges, the total number of triangular faces is $\frac{VT}{3}$, and similarly, the total number of pentagonal faces is $\frac{VP}{5}$. Since each face is either a triangular face or a pentagonal face, we have $\frac{VT}{3}+\frac{VP}{5}=32$. From (1) and (2), we get $3T+5P=16$, whose only non-negative integer solution is $T=P=2$. Consequently, $V=30$, so $P+T+V=34$.