There are 10 points on a circle. By joining them how many chords can be drawn?

Answer

To find the number of chords that can be drawn by joining 10 points on a circle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Chords**: A chord is formed by joining any two points on the circle. Therefore, to find the total number of chords, we need to determine how many ways we can select 2 points from the 10 points available.

2. **Using Combinations**: The number of ways to choose 2 points from 10 can be calculated using the combination formula:
\[ \text{Number of chords} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of points (10 in this case) and \( r \) is the number of points to choose (2 for a chord).

3. **Applying the Formula**: Plugging in the values:
\[ \text{Number of chords} = \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10!}{2! \cdot 8!} \]

4. **Simplifying the Factorials**: We can simplify the factorials:
\[ \binom{10}{2} = \frac{10 \times 9 \times 8!}{2! \times 8!} \]
Here, the \( 8! \) cancels out:
\[ = \frac{10 \times 9}{2!} \]

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

6. **Final Calculation**: Now substituting back:
\[ = \frac{10 \times 9}{2} = \frac{90}{2} = 45 \]

7. **Conclusion**: Therefore, the total number of chords that can be drawn by joining the 10 points on the circle is **45**.