A polygonal pyramid is a three-dimensional solid. Its base is a regular polygon. Each of the vertices of the polygonal base is connected to a single point, called the apex. The sum of the number of edges and the number of vertices of a particular polygonal pyramid is 1915. How many faces does this pyramid have?
Suppose that the base of the pyramid has $n$ sides.

The base will also have $n$ vertices. Since the pyramid has one extra vertex (the apex), then it has $n+1$ vertices in total.

The pyramid has $n+1$ faces: the base plus $n$ triangular faces formed by each edge of the base and the apex.

The pyramid has $2 n$ edges: the $n$ sides that form the base plus one edge joining each of the $n$ vertices of the base to the apex.

From the given information, $2 n+(n+1)=1915$.

Thus, $3 n=1914$ and so $n=638$.

Since the pyramid has $n+1$ faces, then it has 639 faces.

ANSWER: 639