In a party 15 people shake their hands with each other. How many times did the hand-shakes take place?

Options

A. `""^(15)P_(2)`
B. `""^(15)C_(2)`
C. 15!
D. `2xx15!`

Answer

To find the total number of handshakes that occur when 15 people shake hands with each other, we can use the concept of combinations. Each handshake involves a pair of people, and we need to find how many unique pairs can be formed from 15 people.

### Step-by-Step Solution:

1. **Identify the total number of people**:
We have a total of 15 people at the party.

2. **Understand the handshake requirement**:
A handshake occurs between two people. Therefore, we need to choose 2 people from the 15 to form a handshake.

3. **Use the combination formula**:
The number of ways to choose 2 people from 15 is given by the combination formula \( nCk \), where \( n \) is the total number of people and \( k \) is the number of people to choose. In this case, it is \( 15C2 \).

\[ 15C2 = \frac{15!}{(15-2)! \cdot 2!} \]

4. **Calculate the factorials**:
We can simplify this expression:
\[ 15C2 = \frac{15!}{13! \cdot 2!} = \frac{15 \times 14 \times 13!}{13! \cdot 2!} \]

5. **Cancel out the common terms**:
The \( 13! \) cancels out:
\[ 15C2 = \frac{15 \times 14}{2!} \]

6. **Calculate \( 2! \)**:
We know that \( 2! = 2 \):
\[ 15C2 = \frac{15 \times 14}{2} = \frac{210}{2} = 105 \]

7. **Conclusion**:
Therefore, the total number of handshakes that took place is **105**.