In a room of ten people, how many handshakes will there be if every person shakes hands with everyone else exactly once?

To solve the problem of how many handshakes will occur in a room of ten people if each person shakes hands with every other person exactly once, we can use a combinatorial approach.

### Step-by-Step Solution:
1. **Understand the problem**: Each handshake involves two people. Since every person shakes hands with everyone else exactly once, we're interested in finding out how many unique pairs of people can be formed.

2. **Combinatorial approach**: The number of ways to choose 2 people from a group of 10 is given by the combination formula:
   \[
   \text{Number of handshakes} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45
   \]

Thus, there will be **45 handshakes** in total.

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### Related Questions:
1. How would the number of handshakes change if there were 20 people in the room?
2. What is the general formula for calculating the number of handshakes in a room of \( n \) people?
3. How many handshakes would occur if there were 5 people in the room?
4. How does the number of handshakes scale with the number of people?
5. Can you derive the combination formula \(\binom{n}{2}\) from basic principles?
6. What if one person refuses to shake hands? How many handshakes then?
7. What if each person only shakes hands with half of the others?
8. How would the scenario change if people could shake hands more than once?

### Tip:
For problems involving "everyone interacting with everyone else," combinations are often the right tool to count the interactions!

Discover how many handshakes occur in a room of ten people where each person shakes hands with every other person exactly once. Using combinatorial mathematics, learn the combination formula and its application to counting unique pairs.

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