If 7 points out of 12 are in the same straight line, then the number of triangles formed is

Options

A. 19
B. 158
C. 185
D. 201

Answer

To solve the problem of how many triangles can be formed from 12 points where 7 points are collinear, we can follow these steps:

### Step 1: Calculate the total number of ways to choose 3 points from 12 points.
We can use the combination formula \( nCr \), which is given by:

\[ nCr = \frac{n!}{r!(n-r)!} \]

For our case, we want to find \( 12C3 \):

\[ 12C3 = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = \frac{1320}{6} = 220 \]

### Step 2: Calculate the number of ways to choose 3 points from the 7 collinear points.
Since the 7 points are collinear, any combination of these points will not form a triangle. Therefore, we need to calculate \( 7C3 \):

\[ 7C3 = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]

### Step 3: Subtract the collinear combinations from the total combinations.
To find the number of triangles that can actually be formed, we subtract the combinations of collinear points from the total combinations:

\[ \text{Number of triangles} = 12C3 - 7C3 = 220 - 35 = 185 \]

### Conclusion:
The total number of triangles that can be formed from the given points is **185**.

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