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

Options

A. 185
B. 158
C. 172
D. None of these

Answer

To solve the problem of finding the number of triangles that 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.

To form a triangle, we need to select 3 points. The total number of ways to choose 3 points from 12 points is given by the combination formula:

\[ \text{Total triangles} = \binom{12}{3} \]

Using the combination formula:

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

For our case:

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

### Step 2: Calculate the number of ways to choose 3 points from the 7 collinear points.

Since the 7 points are collinear, they cannot form a triangle. Therefore, we need to subtract the number of ways to choose 3 points from these 7 collinear points:

\[ \text{Collinear triangles} = \binom{7}{3} \]

Calculating this gives:

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

### Step 3: Subtract the number of collinear triangles from the total triangles.

Now, we subtract the number of collinear triangles from the total triangles to find the number of valid triangles that can be formed:

\[ \text{Valid triangles} = \binom{12}{3} - \binom{7}{3} = 220 - 35 = 185 \]

### Final Answer:

Thus, the number of triangles that can be formed is **185**.

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