There are 14 points in a plane, out of which 4 are collinear. Find the number of triangles made by these points.

Options

A. 364
B. 360
C. 368
D. 365

Answer

To find the number of triangles that can be formed from 14 points in a plane, where 4 of these points are collinear, we can follow these steps:

### Step 1: Calculate the total number of triangles from 14 points
If all 14 points were non-collinear, the number of ways to choose 3 points from 14 can be calculated using the combination formula:

\[ \text{Number of triangles} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

where \( n \) is the total number of points and \( r \) is the number of points to choose.

For our case:

\[ \binom{14}{3} = \frac{14!}{3!(14-3)!} = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = 364 \]

### Step 2: Calculate the number of triangles that cannot be formed due to collinearity
Since 4 points are collinear, any triangle formed by choosing 3 points from these 4 collinear points will not be a valid triangle. The number of ways to choose 3 points from these 4 collinear points is:

\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = 4 \]

### Step 3: Subtract the invalid triangles from the total triangles
Now, we subtract the number of invalid triangles (formed by the collinear points) from the total number of triangles:

\[ \text{Valid triangles} = \text{Total triangles} - \text{Invalid triangles} \]

\[ \text{Valid triangles} = 364 - 4 = 360 \]

### Conclusion
Thus, the number of triangles that can be formed from the 14 points, where 4 are collinear, is **360**.

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