There an n persons sitting around a circular table. Each person shakes hands with everybody except the person sitting on both sides of him. The total number of hand shakes are 90. Then find n.

Answer

To solve the problem, we need to find the number of persons \( n \) sitting around a circular table, where each person shakes hands with everyone except the two people sitting next to them, and the total number of handshakes is given as 90.

### Step-by-step Solution:

1. **Understanding Handshakes**:
Each person shakes hands with \( n - 3 \) people (since they do not shake hands with themselves and the two adjacent persons).

2. **Total Handshakes Calculation**:
The total number of handshakes can be calculated using the formula:
\[ \text{Total Handshakes} = \frac{n(n - 3)}{2} \]
Here, \( n(n - 3) \) counts the total handshakes (each handshake is counted twice, hence we divide by 2).

3. **Setting up the Equation**:
We know from the problem statement that the total number of handshakes is 90. Therefore, we can set up the equation:
\[ \frac{n(n - 3)}{2} = 90 \]

4. **Multiplying through by 2**:
To eliminate the fraction, multiply both sides by 2:
\[ n(n - 3) = 180 \]

5. **Rearranging the Equation**:
Rearranging gives us a standard quadratic equation:
\[ n^2 - 3n - 180 = 0 \]

6. **Factoring the Quadratic**:
We need to factor the quadratic equation. We look for two numbers that multiply to \(-180\) and add to \(-3\). The numbers are \( -15 \) and \( 12 \):
\[ (n - 15)(n + 12) = 0 \]

7. **Finding the Roots**:
Setting each factor to zero gives:
\[ n - 15 = 0 \quad \Rightarrow \quad n = 15 \]
\[ n + 12 = 0 \quad \Rightarrow \quad n = -12 \quad (\text{not valid since } n \text{ must be positive}) \]

8. **Conclusion**:
The only valid solution is:
\[ n = 15 \]

### Final Answer:
The number of persons sitting around the table is \( n = 15 \).

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