There are 12 points in a plane in which 6 are collinear. Number of different triangles that can be drawn by joining them is

Options

A. 220
B. 200
C. 240
D. 320

Answer

To solve the problem of finding the number of different triangles that can be formed by joining 12 points in a plane where 6 points are collinear, we can follow these steps:

### Step 1: Understand the problem
We have a total of 12 points, out of which 6 are collinear. To form a triangle, we need to select 3 points. However, if all 3 points chosen are collinear, they will not form a triangle.

### Step 2: Calculate the total number of triangles without restrictions
We can calculate the total number of ways to choose 3 points from the 12 points using the combination formula:
\[ \text{Total triangles} = \binom{12}{3} \]
Calculating this gives:
\[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]

### Step 3: Calculate the number of triangles that cannot be formed
Next, we need to find the number of triangles that can be formed using the 6 collinear points. Since these points are collinear, any selection of 3 points from these will not form a triangle:
\[ \text{Collinear triangles} = \binom{6}{3} \]
Calculating this gives:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 4: Subtract the collinear triangles from the total triangles
Now, we subtract the number of collinear triangles from the total number of triangles:
\[ \text{Valid triangles} = \text{Total triangles} - \text{Collinear triangles} = 220 - 20 = 200 \]

### Conclusion
Thus, the number of different triangles that can be drawn by joining these points is:
\[ \boxed{200} \]

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