Question

There are 11 points in a plane of which 5 points are collinear. Then the total number of distinct quadrilaterals with vertices at these points is ______.

• A. 265
• B. 330
• C. 250
• D. 325

Hint:

For polygon formation problems involving collinear points, always ask yourself: "How many points from the collinear set will destroy the shape?" For a quadrilateral, selecting 3 or 4 points from a single line destroys the required 4-sided geometry.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the total number of valid quadrilaterals (4-sided polygons) that can be formed using a given set of 11 points. The primary constraint is that exactly 5 of these points lie on a single straight line.

Step 2: Key Formula or Approach:
A quadrilateral requires choosing exactly 4 non-collinear points.
The easiest approach is "Total minus Invalid":
$$\text{Valid Quadrilaterals} = \binom{\text{Total points}}{4} - (\text{Invalid combinations})$$
An invalid combination occurs if we select 4 points that lie entirely on the collinear line (forming a straight line, not a polygon), or if we select 3 points from the collinear line and 1 point from outside (forming a triangle, not a quadrilateral).

Step 3: Detailed Explanation:
1. Calculate the absolute total combinations of choosing any 4 points from 11:
$$\text{Total} = \binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330$$
2. Calculate invalid case 1: Selecting all 4 points from the 5 collinear points.
$$\text{Invalid}_1 = \binom{5}{4} = 5$$
3. Calculate invalid case 2: Selecting exactly 3 points from the 5 collinear points AND 1 point from the remaining 6 non-collinear points.
$$\text{Invalid}_2 = \binom{5}{3} \times \binom{6}{1} = 10 \times 6 = 60$$
4. Subtract the invalid cases from the total combinations to find the valid quadrilaterals:
$$\text{Valid Quadrilaterals} = 330 - 5 - 60 = 265$$

Step 4: Final Answer:
The total number of distinct quadrilaterals is 265, matching option (a).