here is a group of people in a room, and each of them will shake hands
with every other person once and only once.
(a) If there are 3 people in the room, how many handshakes are there

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 \( \frac{n(n-1)}{2} \).

Here is a group of people in a room, and each of them will shake hands with every other person once and only once. (a) If there are 3 people in the room, how many handshakes are there?

Learn how to calculate the number of handshakes in a group of 3 people using combinatorics. This problem uses the combination formula to find the answer and is suitable for students in Grades 6-8. Explore more examples and discover how handshakes increase as the group grows.

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