seven managers arrive at a board meeting. each manager shakes the hand of all the other managers. how many handshakes take place

This problem can be solved using the concept of combinations. Each handshake involves a unique pair of managers, and we need to find out how many such pairs can be formed from the 7 managers.

The number of ways to choose 2 managers out of 7 for a handshake is given by the combination formula:

\[
\text{Number of handshakes} = \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21
\]

So, there are **21 handshakes** that take place.

Would you like further details or have any other questions?

Here are five related questions:
1. How many handshakes would take place if there were 8 managers?
2. What is the formula for calculating handshakes in general for \( n \) people?
3. How does the number of handshakes change if one manager does not participate?
4. If the managers could shake hands more than once, how would this affect the calculation?
5. How many handshakes would occur if there were 10 managers?

**Tip:** When calculating interactions like handshakes, remember that each interaction is a unique pair, and order does not matter, making it a combination problem.

Learn how to calculate the number of handshakes between seven managers at a board meeting using combinations. Understand the concept of combinations and apply the combination formula to solve for the number of unique handshakes. Suitable for high school students studying combinatorics.

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