8 men entered a lounge simultaneously . if each peron shook hands with the other, then find the total no. of hand shakes.

Options

A. 16
B. 36
C. 56
D. 28

Answer

To find the total number of handshakes among 8 men, we can use the concept of combinations. When each person shakes hands with every other person, we can determine the total number of unique handshakes using the formula for combinations.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 8 men, and each man shakes hands with every other man. We need to find the total number of unique handshakes.

2. **Identifying the Formula**:
The number of ways to choose 2 people from a group of n people (which represents a handshake between two people) is given by the combination formula:
\[ C(n, 2) = \frac{n!}{2!(n-2)!} \]
where \( n \) is the total number of people.

3. **Substituting the Values**:
Here, \( n = 8 \). So we substitute 8 into the formula:
\[ C(8, 2) = \frac{8!}{2!(8-2)!} = \frac{8!}{2! \cdot 6!} \]

4. **Calculating Factorials**:
We can simplify this calculation:
\[ C(8, 2) = \frac{8 \times 7}{2 \times 1} = \frac{56}{2} = 28 \]

5. **Conclusion**:
Therefore, the total number of unique handshakes that occur when 8 men each shake hands with every other man is **28**.

### Final Answer:
The total number of handshakes is **28**.