There are 10 points in a plane, out of these 6 are collinear. The number of triangles formed by joining these points, is

Options

A. `N gt 190`
B. `N le 100`
C. `100 lt N le 140`
D. `140 lt N le 190`

Explanation

To solve the problem of finding the number of triangles that can be formed by joining 10 points in a plane, where 6 of these points are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have a total of 10 points, out of which 6 points are collinear. To form a triangle, we need to select 3 points. However, if all 3 points selected are collinear, they will not form a triangle.

2. **Calculate Total Combinations of Points**:
We can calculate the total number of ways to choose 3 points from the 10 points using the combination formula:
\[ \text{Total combinations} = \binom{10}{3} \]
This can be calculated as:
\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

3. **Calculate Combinations of Collinear Points**:
Next, we need to find the number of ways to choose 3 points from the 6 collinear points, as these will not form a triangle:
\[ \text{Collinear combinations} = \binom{6}{3} \]
This can be calculated as:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

4. **Subtract Collinear Combinations from Total Combinations**:
To find the number of triangles that can actually be formed, we subtract the number of collinear combinations from the total combinations:
\[ \text{Triangles formed} = \binom{10}{3} - \binom{6}{3} = 120 - 20 = 100 \]

5. **Conclusion**:
The total number of triangles that can be formed by joining the given points is 100.

### Final Answer:
The number of triangles formed by joining these points is **100**.