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 triangles formed by joining them.

Options

A. `n!-m!`
B. `^mC_3-^nC_3`
C. `(n-m)C_3-^mC_3`
D. `^nC_3-^mC_3`

Answer

To solve the problem of finding the maximum number of triangles that can be formed by joining n points on a plane, where m of those points are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Points**: We have a total of n points on the plane. Among these, m points are collinear.

2. **Triangles from n Points**: If there were no collinear points, the maximum number of triangles that could be formed by choosing any three points from these n points is given by the combination formula \( \binom{n}{3} \). This is calculated as:
\[ \binom{n}{3} = \frac{n!}{3!(n-3)!} \]

3. **Triangles from Collinear Points**: However, since m points are collinear, any triangle formed by selecting three points from these m collinear points will not actually form a triangle. Thus, the number of ways to choose three points from these m collinear points is given by \( \binom{m}{3} \):
\[ \binom{m}{3} = \frac{m!}{3!(m-3)!} \]

4. **Actual Triangles Formed**: To find the actual number of triangles that can be formed, we need to subtract the number of non-triangle combinations (from collinear points) from the total combinations of triangles (from all points). Therefore, the maximum number of triangles formed is:
\[ \text{Maximum Triangles} = \binom{n}{3} - \binom{m}{3} \]

5. **Final Answer**: Thus, the answer to the question is:
\[ \text{Maximum number of triangles} = \binom{n}{3} - \binom{m}{3} \]

### Conclusion:
The final answer is \( \binom{n}{3} - \binom{m}{3} \).