At the end of a banquet 10 people shake hands with each other. How many handshakes will there be intotal?

To determine the total number of handshakes, we can use the concept of combinations. Each handshake involves a unique pair of people.

Step 1: Identify the total number of people. There are 10 people.

Step 2: Use the combination formula to find the number of ways to choose 2 people out of 10. The combination formula is given by: $C(n, k) = \frac{n!}{k!(n-k)!}$ where $n$ is the total number of items, $k$ is the number of items to choose, and $!$ denotes factorial.

Step 3: Substitute the values into the formula. Here, $n = 10$ and $k = 2$: $C(10, 2) = \frac{10!}{2!(10-2)!} = \frac{10!}{2! \cdot 8!}$

Step 4: Simplify the factorials. $10! = 10 \times 9 \times 8!$ $\frac{10!}{2! \cdot 8!} = \frac{10 \times 9 \times 8!}{2! \times 8!}$

Step 5: Cancel out the common $8!$ terms. $\frac{10 \times 9}{2!} = \frac{10 \times 9}{2 \times 1} = \frac{90}{2} = 45$

Therefore, the total number of handshakes is 45.