There are 7 non collinear points in a plane. By joining these points in pairs, how many distinct triangles can be formed?

Introduction / Context:
This question tests counting of geometric shapes using combinations. When no three points are collinear, every set of three points forms a unique triangle. This leads directly to a combination based formula, which is a standard result in combinatorial geometry.

Given Data / Assumptions:

• There are 7 points in a plane.
• No three points lie on the same straight line (non collinear condition).
• Each triangle is determined by 3 distinct points.
• Triangles are considered distinct based on their vertex sets.

Concept / Approach:
If no three points are collinear, then any selection of 3 distinct points gives a valid triangle. Therefore, the number of triangles equals the number of combinations of 7 points taken 3 at a time, which is 7C3. We simply compute this combination.

Step-by-Step Solution:
Step 1: The number of triangles is the number of ways to choose 3 points out of 7.Step 2: Use the combination formula 7C3 = 7! / (3! * 4!).Step 3: Compute 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1.Step 4: Use cancellation: 7! / (3! * 4!) simplifies to (7 * 6 * 5) / (3 * 2 * 1) = 210 / 6.Step 5: Evaluate 210 / 6 = 35.

Verification / Alternative check:
A quick check is to consider smaller numbers of points, such as 4 non collinear points. In that case, 4C3 = 4 triangles, which you can easily visualise. Extending to 7 points and using the same formula gives 7C3 = 35, which is consistent with this general pattern and uses the same underlying combinatorial reasoning.

Why Other Options Are Wrong:

• 10: This equals 5C2, not 7C3, and undercounts greatly.
• 30: This might come from an incorrect formula or a calculation slip like 6 * 5 instead of 7C3.
• 60: This is too large and could result from using permutations instead of combinations.

Common Pitfalls:
Students sometimes forget the non collinear condition and try to subtract extra cases or worry about degenerate triangles. Another mistake is to use permutations 7P3, which would count ordered triples and not triangles. Always recall that a triangle is determined only by the set of 3 vertices, so combinations are appropriate.

Final Answer:
The number of distinct triangles that can be formed is 35.