13. A dodecahedron is a regular solid with 12 pentagonal faces. A diagonal of a solid is a segment whose endpoints are two vertices of the solid that do not belong to the same face. How many diagonals does a dodecahedron have?
13. The answer is 100. It is necessary to count the pairs (unordered) of vertices not belonging to the same face. At each vertex, 3 faces meet, and in each of these faces, there are 2 vertices that are not on a face that also contains the initial vertex (total 6) plus 2 shared with another face (which are counted twice and are actually a total of 3) plus the vertex itself: so we need to exclude from the count 10 vertices that do not form diagonals. But the vertices of the dodecahedron are $12 \cdot 5 / 3=20$ (12 pentagonal faces, but they meet 3 at a time), so for each of these, we need to consider $20-10=10$ possible vertices as the other end of a diagonal. In total, the diagonals are $20 \cdot 10 / 2=100$, where we have divided by 2 because, by multiplying, each diagonal is counted twice.