Find the number of triangle formed by joining 12 different points on a plane , no three of them being collinear ( with the exeption of 4 points which are collinear ).

Options

A. 126
B. 216
C. 220
D. 222

Answer

To solve the problem of finding the number of triangles that can be formed by joining 12 different points on a plane, where no three points are collinear except for 4 collinear points, we can follow these steps:

### Step 1: Calculate the total number of triangles from 12 points
We can choose any 3 points from the 12 points to form a triangle. The number of ways to choose 3 points from 12 is given by the combination formula \( nCk \), which is calculated as:

\[ 12C3 = \frac{12!}{3!(12-3)!} = \frac{12!}{3! \cdot 9!} \]

### Step 2: Simplify the combination
We can simplify \( 12C3 \):

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

So, the total number of triangles that can be formed using any 3 of the 12 points is 220.

### Step 3: Calculate the number of triangles that cannot be formed from the collinear points
Since 4 of the points are collinear, we cannot form a triangle using these 4 points. The number of ways to choose 3 points from these 4 collinear points is:

\[ 4C3 = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} = 4 \]

### Step 4: Subtract the collinear triangles from the total triangles
Now, we need to subtract the triangles that cannot be formed (the collinear triangles) from the total triangles:

\[ \text{Total triangles} = 220 - 4 = 216 \]

### Final Answer
Thus, the number of triangles that can be formed is **216**.

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