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 Concept:
To form a quadrilateral, we need 4 points. However, if 3 or more points chosen are collinear, they cannot form a quadrilateral.

Step 2: Formula Application:
Total ways to pick 4 points from 11 $= {}^{11}C_4$. Subtract invalid cases: 1. All 4 points from the 5 collinear points: ${}^5C_4$. 2. 3 points from the 5 collinear points and 1 from the remaining 6: ${}^5C_3 \times {}^6C_1$.

Step 3: Explanation:
${}^{11}C_4 = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330$. Invalid Case 1: ${}^5C_4 = 5$. Invalid Case 2: ${}^5C_3 \times {}^6C_1 = 10 \times 6 = 60$. Total quadrilaterals $= 330 - (5 + 60) = 330 - 65 = 265$.

Step 4: Final Answer:
The total number of distinct quadrilaterals is 265.