Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$.  A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon.  At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet.  What is the value of $100P + 10T + V$?
1. **Using Euler's Formula**: Euler's formula for a convex polyhedron states that:
   \[
   V - E + F = 2
   \]
   Given that the polyhedron has \( F = 32 \) faces, we can rewrite Euler's formula as:
   \[
   V - E + 32 = 2 \implies V - E = -30
   \]

2. **Relating Edges and Vertices**: Each vertex is joined by \( T + P \) faces, and thus by \( T + P \) edges. Since each edge is counted twice (once at each end), the total number of edges \( E \) is:
   \[
   E = \frac{V(T + P)}{2}
   \]
   Substituting \( E \) into the modified Euler's formula \( V - E = -30 \):
   \[
   V - \frac{V(T + P)}{2} = -30
   \]
   Simplifying, we get:
   \[
   V \left(1 - \frac{T + P}{2}\right) = -30 \implies V \left(\frac{2 - (T + P)}{2}\right) = -30 \implies V \left(\frac{2 - T - P}{2}\right) = -30
   \]
   Solving for \( V \):
   \[
   V = \frac{60}{T + P - 2}
   \]

3. **Counting Faces**: Each vertex connects \( P \) pentagons and \( T \) triangles. Since each pentagon has 5 vertices and each triangle has 3 vertices, the total number of vertices counted by faces is:
   \[
   V \left(\frac{P}{5} + \frac{T}{3}\right) = 32
   \]
   Substituting \( V = \frac{60}{T + P - 2} \) into the equation:
   \[
   \frac{60}{T + P - 2} \left(\frac{P}{5} + \frac{T}{3}\right) = 32
   \]
   Simplifying:
   \[
   60 \left(\frac{P}{5} + \frac{T}{3}\right) = 32 (T + P - 2)
   \]
   \[
   60 \left(\frac{3P + 5T}{15}\right) = 32 (T + P - 2)
   \]
   \[
   12 (3P + 5T) = 32 (T + P - 2)
   \]
   \[
   36P + 60T = 32T + 32P - 64
   \]
   \[
   4P + 28T = 64
   \]
   \[
   P + 7T = 16
   \]

4. **Solving for \( P \) and \( T \)**: The equation \( P + 7T = 16 \) must be solved for positive integers \( P \) and \( T \). Testing values, we find:
   \[
   T = 2 \implies P + 14 = 16 \implies P = 2
   \]

5. **Calculating \( V \)**: Substituting \( T = 2 \) and \( P = 2 \) into \( V = \frac{60}{T + P - 2} \):
   \[
   V = \frac{60}{2 + 2 - 2} = \frac{60}{2} = 30
   \]

6. **Final Calculation**: We need to find \( 100P + 10T + V \):
   \[
   100P + 10T + V = 100 \cdot 2 + 10 \cdot 2 + 30 = 200 + 20 + 30 = 250
   \]

The final answer is \(\boxed{250}\)