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

Options

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

Answer

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 1: Understanding the Problem
We have a total of 10 points, out of which 6 points are collinear. When points are collinear, they cannot form a triangle. Therefore, we need to calculate the total number of triangles that can be formed from these points and then subtract the triangles that can be formed using the collinear points.

### Step 2: Calculate Total Combinations of 3 Points
To find the total number of triangles that can be formed from 10 points, we use the combination formula \( nCk \), which represents the number of ways to choose \( k \) elements from \( n \) elements without regard to the order of selection.

Here, we need to calculate \( 10C3 \):
\[ 10C3 = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

### Step 3: Calculate Combinations of Collinear Points
Next, we calculate the number of triangles that can be formed using the 6 collinear points. Since these points are collinear, they cannot form a triangle. Thus, we calculate \( 6C3 \):
\[ 6C3 = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 4: Subtract Collinear Combinations from Total Combinations
Now, we subtract the number of triangles that can be formed from the collinear points from the total number of triangles:
\[ N = 10C3 - 6C3 = 120 - 20 = 100 \]

### Conclusion
The number of triangles that can be formed by joining these points is \( N = 100 \).