Question

Out of 9 points in a plane, no three points are in same straight line except four points which are collinear. Then, the number of quadrilaterals which can be formed by joining them, is

• A. 100
• B. 105
• C. 102
• D. 103

Answer 1

Answer: (B)

105

Answer 2

To form a quadrilateral, we need to select 4 points such that no three of them are collinear. We have 9 points in total, with 4 collinear points. This leaves 5 non-collinear points.

We can form quadrilaterals by considering the following cases:

1. Choosing 4 points from the 5 non-collinear points: $\binom{5}{4} = 5$ ways.
2. Choosing 3 points from the 5 non-collinear points and 1 point from the 4 collinear points: $\binom{5}{3} \times \binom{4}{1} = 10 \times 4 = 40$ ways.
3. Choosing 2 points from the 5 non-collinear points and 2 points from the 4 collinear points: $\binom{5}{2} \times \binom{4}{2} = 10 \times 6 = 60$ ways.

The case of choosing 1 point from the 5 non-collinear points and 3 points from the 4 collinear points is invalid because it would result in 3 collinear points among the chosen 4.

Total number of quadrilaterals = $5 + 40 + 60 = 105$.