In this problem, we need to determine how many handshakes occur if each person in a group shakes hands with every other person exactly once.

For part (a), if there are 3 people in the room, each person will shake hands with every other person. We can use the combination formula to find the number of distinct handshakes:

$\text{Number of handshakes} = \binom{n}{2} = \frac{n(n-1)}{2}$

Where:

• $n$ is the number of people in the room.
• $\binom{n}{2}$ is the number of ways to choose 2 people from $n$ to shake hands.

For $n = 3$:

$\text{Number of handshakes} = \binom{3}{2} = \frac{3(3-1)}{2} = \frac{3 \times 2}{2} = 3$

So, if there are 3 people in the room, there will be 3 handshakes.

Would you like details on this calculation or have further questions?

Here are 5 related questions:

1. How many handshakes are there if 4 people are in the room?
2. What is the general formula for the number of handshakes for any $n$ people?
3. How does the number of handshakes grow as the number of people increases?
4. Can you visualize the handshake problem using a graph or diagram?
5. What happens if the group size is larger, like 10 people?

Tip: To calculate the number of ways two people can be chosen from a group of $n$, always use the combination formula $\binom{n}{2}$, which simplifies to $\frac{n(n-1)}{2}$.