When ten persons shake hands with one another in how many ways it is possible ?

Options

A. 20
B. 25
C. 40
D. 45

Answer

To solve the problem of how many ways 10 persons can shake hands with one another, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to determine how many unique handshakes can occur when 10 people each shake hands with every other person.

2. **Identifying the Combination**:
A handshake involves 2 people. Therefore, we need to find the number of ways to choose 2 people from a group of 10. This can be represented using combinations.

3. **Using the Combination Formula**:
The formula for combinations is given by:
\[ nCr = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items (people), \( r \) is the number of items to choose (2 for a handshake), and \( ! \) denotes factorial.

4. **Applying the Formula**:
For our problem, we have \( n = 10 \) and \( r = 2 \):
\[ 10C2 = \frac{10!}{2!(10-2)!} = \frac{10!}{2! \cdot 8!} \]

5. **Calculating Factorials**:
We can simplify this expression:
\[ 10! = 10 \times 9 \times 8! \]
Therefore,
\[ 10C2 = \frac{10 \times 9 \times 8!}{2! \times 8!} \]
The \( 8! \) cancels out:
\[ 10C2 = \frac{10 \times 9}{2!} \]

6. **Calculating \( 2! \)**:
We know that \( 2! = 2 \times 1 = 2 \). Thus, we can substitute this into our equation:
\[ 10C2 = \frac{10 \times 9}{2} = \frac{90}{2} = 45 \]

7. **Conclusion**:
Therefore, the total number of ways in which 10 persons can shake hands with one another is **45**.