There are 2n points in a plane in which m are collinear. Number of quadrilaterals formed by joining these lines: a. is equal to `.^(2n)C_(4)-.^(m)C_(4)` b. is greater than `.^(2n)C_(4)_.^(m)C_(4)` c. is less than `.^(2n)C_(4)-.^(m)C_(4)` d. None of these

Options

A. is equal to `.^(2n)C_(4)-.^(m)C_(4)`
B. is greater than `.^(2n)C_(4)_.^(m)C_(4)`
C. is less than `.^(2n)C_(4)-.^(m)C_(4)`
D. None of these

Answer

To solve the problem of finding the number of quadrilaterals that can be formed from 2n points in a plane, where m points are collinear, we can follow these steps:

### Step 1: Calculate the total number of ways to choose 4 points from 2n points.
The total number of ways to choose 4 points from 2n points is given by the combination formula:
\[ \binom{2n}{4} \]

### Step 2: Identify the collinear points.
Since m points are collinear, we need to consider the cases where these collinear points could affect the formation of quadrilaterals.

### Step 3: Subtract the cases where all 4 points are collinear.
If we choose 4 points from the m collinear points, they cannot form a quadrilateral. The number of ways to choose 4 points from these m collinear points is:
\[ \binom{m}{4} \]
Thus, we subtract this from our total.

### Step 4: Subtract the cases where 3 points are collinear and 1 point is non-collinear.
If we select 3 collinear points and 1 non-collinear point, they also cannot form a quadrilateral. The number of ways to choose 3 collinear points from m and 1 point from the remaining (2n - m) non-collinear points is:
\[ \binom{m}{3} \cdot \binom{2n - m}{1} \]
We need to subtract this from our total as well.

### Step 5: Combine the results.
The total number of valid quadrilaterals can be expressed as:
\[ \text{Number of quadrilaterals} = \binom{2n}{4} - \binom{m}{4} - \left( \binom{m}{3} \cdot \binom{2n - m}{1} \right) \]

### Conclusion
Now we can analyze the options provided in the question:
- The number of quadrilaterals formed is less than \( \binom{2n}{4} - \binom{m}{4} \) because we have subtracted additional cases where 3 collinear points are chosen.

Thus, the correct answer is:
**c. is less than \( \binom{2n}{4} - \binom{m}{4} \)**.