Combinatorics

To find the number of triangles with vertices on different sides of the convex quadrilateral PQRS, we need to consider points on each side of the quadrilateral. Analyzing what is given:

• 3 points on side PQ
• 4 points on side QR
• 5 points on side RS
• 6 points on side PS

To form a triangle, we need to select one point from each of the three different sides of the quadrilateral. Therefore, we need to compute the possible combinations of selecting points from any trio of these sides.

The sides can be categorized into groups of three in the following combinations:

1. Select one point from side PQ, one point from side QR, and one point from side RS.
2. Select one point from side PQ, one point from side RS, and one point from side PS.
3. Select one point from side PQ, one point from side PS, and one point from side QR.
4. Select one point from side QR, one point from side RS, and one point from side PS.

Adding all these computations together gives the total number of triangles possible:

Thus, the correct answer is 342.