Question

S is a set of points in the plane. How many distinct triangles can be drawn that have three of the points in S as vertices?
(1) The number of distinct points in S is 5.
(2) No three of the points in S are collinear.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is C Both statements together are sufficient, but neither statement alone is sufficient.

1. Given that the number of points in S is 5, the number of triangles can be 0 (if the points are collinear) and the number of triangles can be greater than 0 (if the points are not all collinear); NOT sufficient.
2. Given that no three points of S are collinear, the number of triangles can be 1 (if S consists of 3 points) and the number of triangles can be 4 (if S consists of 4 points); NOT sufficient.
   
3. Taking (1) and (2) together, the number of distinct triangles must be which is the number of combinations of 5 points taken 3 at a time.

• The correct answer is C; both statements together are sufficient.