There are 10 non collinear points in a plane. By joining them how many triangles can be made?

Answer

To find the number of triangles that can be formed by joining 10 non-collinear points in a plane, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Requirement for a Triangle**:
To form a triangle, we need to select 3 points from the given points. Since the points are non-collinear, any selection of 3 points will always form a triangle.

2. **Identify the Total Number of Points**:
We have a total of 10 points.

3. **Use the Combination Formula**:
The number of ways to choose 3 points from 10 can be calculated using the combination formula:
\[ nCr = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of points (10 in this case) and \( r \) is the number of points to choose (3 for a triangle).

4. **Substitute the Values into the Formula**:
Here, \( n = 10 \) and \( r = 3 \):
\[ 10C3 = \frac{10!}{3!(10-3)!} = \frac{10!}{3! \cdot 7!} \]

5. **Simplify the Factorials**:
We can expand \( 10! \) and cancel \( 7! \):
\[ 10! = 10 \times 9 \times 8 \times 7! \]
So,
\[ 10C3 = \frac{10 \times 9 \times 8 \times 7!}{3! \times 7!} = \frac{10 \times 9 \times 8}{3!} \]

6. **Calculate \( 3! \)**:
\( 3! = 3 \times 2 \times 1 = 6 \).

7. **Final Calculation**:
Now substituting \( 3! \) back into the equation:
\[ 10C3 = \frac{10 \times 9 \times 8}{6} \]
Calculate the numerator:
\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]
Now divide by 6:
\[ \frac{720}{6} = 120 \]

8. **Conclusion**:
Therefore, the number of triangles that can be formed by joining 10 non-collinear points is **120**.