There are n points in a plane no three of which are in the same straight line with the exception of m(`n>m`)`points which are all in the same straight line. Find the number of triangles formed by joining them.

Options

A. `""^(n-m)C_3`
B. `""^(n)C_3 -""^(m)C_3`
C. `""^(n)C_m`
D. none of (a),(b ),(c )

Answer

To solve the problem of finding the number of triangles formed by joining n points in a plane, where no three points are collinear except for m points that are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Points**:
We have a total of n points in the plane. Out of these, m points are collinear, meaning they lie on the same straight line.

2. **Calculate Total Triangles from n Points**:
To find the total number of triangles that can be formed by selecting any three points from the n points, we use the combination formula:
\[ \text{Total triangles} = \binom{n}{3} = \frac{n(n-1)(n-2)}{6} \]

3. **Calculate Collinear Triangles from m Points**:
Since the m points are collinear, any triangle formed by selecting three points from these m points will not be a valid triangle. The number of ways to choose 3 points from these m collinear points is:
\[ \text{Collinear triangles} = \binom{m}{3} = \frac{m(m-1)(m-2)}{6} \]

4. **Subtract Collinear Triangles from Total Triangles**:
The valid triangles are those formed by the n points minus the invalid triangles formed by the m collinear points. Thus, the number of valid triangles is:
\[ \text{Valid triangles} = \binom{n}{3} - \binom{m}{3} \]

5. **Final Result**:
Therefore, the final expression for the number of triangles that can be formed is:
\[ \text{Number of triangles} = \binom{n}{3} - \binom{m}{3} \]