There are 18 points in a plane such that no three of them are in the same line except five points which are collinear. The number of triangles formed by these points, is

Options

A. 805
B. 806
C. 816
D. none of these

Answer

To solve the problem of finding the number of triangles that can be formed from 18 points in a plane, where no three points are collinear except for five points that are collinear, we can follow these steps:

### Step 1: Understand the Points Configuration
We have a total of 18 points:
- 5 points are collinear (let's call them A1, A2, A3, A4, A5).
- 13 points are non-collinear (let's call them B1, B2, ..., B13).

### Step 2: Identify Cases for Triangle Formation
To form a triangle, we need to select 3 points. We will consider three cases based on the selection of collinear and non-collinear points.

#### Case 1: 2 points from collinear points and 1 from non-collinear points
- We can choose 2 points from the 5 collinear points and 1 point from the 13 non-collinear points.
- The number of ways to choose 2 points from 5 is given by \( \binom{5}{2} \).
- The number of ways to choose 1 point from 13 is given by \( \binom{13}{1} \).

Calculating this:
\[ \text{Number of triangles} = \binom{5}{2} \times \binom{13}{1} = 10 \times 13 = 130 \]

#### Case 2: 1 point from collinear points and 2 from non-collinear points
- We can choose 1 point from the 5 collinear points and 2 points from the 13 non-collinear points.
- The number of ways to choose 1 point from 5 is \( \binom{5}{1} \).
- The number of ways to choose 2 points from 13 is \( \binom{13}{2} \).

Calculating this:
\[ \text{Number of triangles} = \binom{5}{1} \times \binom{13}{2} = 5 \times 78 = 390 \]

#### Case 3: All points from non-collinear points
- We can choose all 3 points from the 13 non-collinear points.
- The number of ways to choose 3 points from 13 is \( \binom{13}{3} \).

Calculating this:
\[ \text{Number of triangles} = \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \]

### Step 3: Total Number of Triangles
Now, we add the number of triangles from all three cases:
\[ \text{Total triangles} = 130 + 390 + 286 = 806 \]

### Final Answer
The total number of triangles that can be formed by these points is **806**.