Handshake Combinations Calculation

This problem involves finding the number of unique pairs possible from a group of 6 people, where each pair represents a handshake. Since shaking hands is mutual (Person A shaking Person B's hand is the same handshake as Person B shaking Person A's hand), the order does not matter. This indicates that we should use combinations.

Understanding Combinations

The problem asks for the number of ways to choose 2 people from a group of 6, without regard to the order of selection. This is a classic combination problem.

The formula for combinations is given by:

Where:

• $n$ is the total number of items (people in this case).
• $k$ is the number of items to choose for each combination (2 people for a handshake).
• $!$ denotes the factorial (e.g., $4! = 4 \times 3 \times 2 \times 1$).

Calculating Handshakes

In this problem, we have $n = 6$ people and we are choosing $k = 2$ people for each handshake.

Applying the combination formula:

Now, let's calculate the factorials:

• $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
• $2! = 2 \times 1 = 2$
• $4! = 4 \times 3 \times 2 \times 1 = 24$

Substitute these values back into the formula:

Result

There are 15 different possible handshakes among 6 people.