There are 10 points in a plane out of which 5 are collinear. The number of triangles that can be drawn will be

Options

A. 120
B. 110
C. 100
D. 78

Answer

To find the number of triangles that can be formed from 10 points in a plane, where 5 of those points are collinear, we can follow these steps:

### Step 1: Calculate the total number of triangles that can be formed from 10 points.
To form a triangle, we need to choose 3 points from the available points. The total number of ways to choose 3 points from 10 is given by the combination formula:

\[ \text{Total triangles} = \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

### Step 2: Calculate the number of triangles that cannot be formed due to collinearity.
Since 5 points are collinear, any triangle formed using these 5 points will not be valid (as they lie on a straight line). The number of ways to choose 3 points from these 5 collinear points is:

\[ \text{Collinear triangles} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Subtract the number of invalid triangles from the total triangles.
Now, we subtract the number of triangles that cannot be formed from the total number of triangles:

\[ \text{Valid triangles} = \text{Total triangles} - \text{Collinear triangles} = 120 - 10 = 110 \]

### Final Answer:
The number of triangles that can be drawn from the given points is **110**.

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