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

Options

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

Answer

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

### Step 1: Understand the problem
We have a total of 10 points, and we need to find how many triangles can be formed by selecting 3 points from these. However, we must account for the fact that 6 of these points are collinear, meaning they cannot form a triangle.

### Step 2: Calculate the total number of triangles
The total number of ways to choose 3 points from 10 points is given by the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \). Here, \( n = 10 \) and \( r = 3 \).

\[ C(10, 3) = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

### Step 3: Calculate the triangles formed by collinear points
Next, we need to calculate how many triangles can be formed using the 6 collinear points. Since collinear points cannot form a triangle, we need to subtract the number of ways to choose 3 points from these 6 points.

\[ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 4: Subtract the collinear combinations from the total
Now, we subtract the number of triangles that can be formed from the collinear points from the total number of triangles.

\[ N = C(10, 3) - C(6, 3) = 120 - 20 = 100 \]

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

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