In a party, every person shakes his hand with every other person only once. If total number of handshakes is 210, then find the number of persons.

Answer

To solve the problem of finding the number of persons at a party where each person shakes hands with every other person exactly once, we can use the concept of combinations.

### Step-by-Step Solution:

1. **Understanding the Problem**:
Each handshake involves two people. If there are \( n \) persons at the party, the total number of unique handshakes can be represented as the combination of \( n \) taken 2 at a time, denoted as \( C(n, 2) \).

2. **Using the Combination Formula**:
The formula for combinations is given by:
\[ C(n, 2) = \frac{n(n-1)}{2} \]
This formula calculates the number of ways to choose 2 people from \( n \) persons.

3. **Setting Up the Equation**:
According to the problem, the total number of handshakes is 210. Therefore, we can set up the equation:
\[ \frac{n(n-1)}{2} = 210 \]

4. **Multiplying Both Sides by 2**:
To eliminate the fraction, we multiply both sides of the equation by 2:
\[ n(n-1) = 420 \]

5. **Rearranging the Equation**:
We can rearrange this into a standard quadratic equation:
\[ n^2 - n - 420 = 0 \]

6. **Applying the Quadratic Formula**:
The quadratic formula is given by:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = -420 \). Plugging in these values:
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-420)}}{2 \cdot 1} \]
\[ n = \frac{1 \pm \sqrt{1 + 1680}}{2} \]
\[ n = \frac{1 \pm \sqrt{1681}}{2} \]

7. **Calculating the Square Root**:
The square root of 1681 is 41:
\[ n = \frac{1 \pm 41}{2} \]

8. **Finding the Values of n**:
This gives us two potential solutions:
\[ n = \frac{42}{2} = 21 \quad \text{and} \quad n = \frac{-40}{2} = -20 \]
Since the number of persons cannot be negative, we take \( n = 21 \).

9. **Conclusion**:
The number of persons at the party is \( \boxed{21} \).