Answer these questions based on the following informations
There are n points on a plane, out of which m are collinear points, such that `mltn`
Find the maximum number of quadrilaterals formed by joining them.

Options

A. `^mC_4+^nC_4`
B. `^nC_4+^mC_4`
C. `^nC_4-^mC_4`
D. `^nC_4-^mC_4+1`

Answer

To find the maximum number of quadrilaterals that can be formed by joining n points on a plane, out of which m are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a total of n points.
- Out of these, m points are collinear. Collinear points cannot form a quadrilateral since they lie on the same line.

2. **Calculating Total Quadrilaterals without Collinearity**:
- If all n points were non-collinear, the number of ways to choose 4 points from n points to form a quadrilateral is given by the combination formula:
\[ \text{Total quadrilaterals} = \binom{n}{4} = \frac{n!}{4!(n-4)!} \]

3. **Calculating Quadrilaterals from Collinear Points**:
- Since m points are collinear, we cannot form a quadrilateral using these m points. The number of ways to choose 4 points from these m collinear points is:
\[ \text{Collinear quadrilaterals} = \binom{m}{4} = \frac{m!}{4!(m-4)!} \]
- These combinations do not contribute to the formation of a quadrilateral.

4. **Finding the Maximum Number of Quadrilaterals**:
- To find the maximum number of quadrilaterals that can be formed, we subtract the number of collinear combinations from the total combinations:
\[ \text{Maximum quadrilaterals} = \binom{n}{4} - \binom{m}{4} \]

5. **Final Answer**:
- Therefore, the maximum number of quadrilaterals formed by joining n points, out of which m are collinear, is:
\[ \text{Answer} = \binom{n}{4} - \binom{m}{4} \]