How many triangles can be drawn through n points on a circle ?

Answer

To determine how many triangles can be drawn through \( n \) points on a circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Number of Points Needed**:
To form a triangle, we need to select 3 points.

2. **Determine the Total Number of Points**:
We are given \( n \) points on the circle.

3. **Use the Combination Formula**:
The number of ways to choose 3 points from \( n \) points is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( r \) is the number of points to choose (in this case, \( r = 3 \)).

4. **Substitute the Values**:
We substitute \( r = 3 \) into the combination formula:
\[ \binom{n}{3} = \frac{n!}{3!(n-3)!} \]

5. **Simplify the Expression**:
This can be simplified to:
\[ \binom{n}{3} = \frac{n(n-1)(n-2)}{3!} = \frac{n(n-1)(n-2)}{6} \]

6. **Conclusion**:
Therefore, the total number of triangles that can be formed using \( n \) points on a circle is:
\[ \frac{n(n-1)(n-2)}{6} \]

### Final Answer:
The number of triangles that can be drawn through \( n \) points on a circle is \( \frac{n(n-1)(n-2)}{6} \).