How many triangles may be formed by joining any three of the nine points when
no three of them are collinear ,

Answer

To find the number of triangles that can be formed by joining any three of the nine points when no three of them are collinear, we can use the concept of combinations.

### Step-by-Step Solution:

1. **Understand the Problem**: We need to determine how many triangles can be formed by selecting any three points from a total of nine points, given that no three points are collinear.

2. **Use the Combination Formula**: The number of ways to choose \( r \) objects from \( n \) objects is given by the combination formula \( nCr \), which is calculated as:
\[ nCr = \frac{n!}{r!(n-r)!} \]
In this case, we want to choose 3 points from 9, so we will use \( 9C3 \).

3. **Substitute Values into the Formula**: Here, \( n = 9 \) and \( r = 3 \):
\[ 9C3 = \frac{9!}{3!(9-3)!} = \frac{9!}{3! \cdot 6!} \]

4. **Simplify the Factorials**: We can simplify \( 9! \) as follows:
\[ 9! = 9 \times 8 \times 7 \times 6! \]
Therefore, substituting this back into the equation gives:
\[ 9C3 = \frac{9 \times 8 \times 7 \times 6!}{3! \cdot 6!} \]
The \( 6! \) cancels out:
\[ 9C3 = \frac{9 \times 8 \times 7}{3!} \]

5. **Calculate \( 3! \)**: The value of \( 3! \) is:
\[ 3! = 3 \times 2 \times 1 = 6 \]

6. **Final Calculation**: Now substitute \( 3! \) back into the equation:
\[ 9C3 = \frac{9 \times 8 \times 7}{6} \]
Calculate the numerator:
\[ 9 \times 8 = 72 \]
\[ 72 \times 7 = 504 \]
Now divide by 6:
\[ 9C3 = \frac{504}{6} = 84 \]

7. **Conclusion**: Therefore, the number of triangles that can be formed by joining any three of the nine points is \( \boxed{84} \).