There are 12 points in a plane of which 5 are collinear.
The number of triangles that can be formed with vertices at thest points is……

Answer

To solve the problem of finding the number of triangles that can be formed from 12 points in a plane, of which 5 are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have a total of 12 points, and among these, 5 points are collinear. A triangle cannot be formed using collinear points since they lie on the same line.

2. **Calculating Total Combinations**: First, we calculate the total number of ways to choose 3 points from the 12 points. This can be done using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of points and \( r \) is the number of points we want to choose.

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

Using the formula for combinations:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

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

3. **Calculating Collinear Combinations**: Next, we need to subtract the combinations that do not form a triangle. Since there are 5 collinear points, we calculate the number of ways to choose 3 points from these 5 collinear points:

\[ \text{Collinear combinations} = \binom{5}{3} \]

Again using the combination formula:
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

4. **Calculating Valid Triangles**: Finally, we subtract the collinear combinations from the total combinations to find the number of valid triangles:

\[ \text{Valid triangles} = \text{Total combinations} - \text{Collinear combinations} \]
\[ \text{Valid triangles} = 220 - 10 = 210 \]

### Final Answer:
The number of triangles that can be formed with vertices at these points is **210**.

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