There are twelve points on a plane, out of which four are collinear points
Find the maximum number of quadrilaterals formed by joining the twelve points.

Options

A. (A) 495
B. (B) 491
C. (C) 494
D. (D) 400

Answer

To find the maximum number of quadrilaterals that can be formed by joining twelve points on a plane, where four of the points are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 12 points in total, and among these, 4 points are collinear. A quadrilateral can be formed by selecting any 4 points, but if all 4 points selected are collinear, they cannot form a quadrilateral.

2. **Calculating Total Combinations**:
The total number of ways to choose 4 points from 12 points can be calculated using the combination formula:
\[ \text{Total combinations} = \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} \]
Simplifying this gives:
\[ \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

3. **Subtracting Invalid Combinations**:
We need to subtract the combinations where all 4 points are collinear. The number of ways to choose 4 points from the 4 collinear points is:
\[ \binom{4}{4} = 1 \]
This means that there is only 1 way to choose all 4 collinear points, which does not form a quadrilateral.

4. **Final Calculation**:
Therefore, the maximum number of quadrilaterals that can be formed is:
\[ \text{Valid quadrilaterals} = \text{Total combinations} - \text{Invalid combinations} = 495 - 1 = 494 \]

5. **Conclusion**:
Thus, the maximum number of quadrilaterals that can be formed by joining the twelve points is **494**.