Arkady, Boris, Vladimir, Grigory and Dmitry shook hands when they met (each shook hands with each other once). How many handshakes were done

a. 5
b. 10
c. 15
d. 25

This is a problem of counting handshakes, which can be solved using combinations.

If there are \( n \) people, each shaking hands with every other person once, the total number of handshakes is given by the combination formula:

\[
\text{Total handshakes} = \binom{n}{2} = \frac{n(n-1)}{2}
\]

For this problem, there are 5 people (Arkady, Boris, Vladimir, Grigory, and Dmitry), so:

\[
\text{Total handshakes} = \binom{5}{2} = \frac{5(5-1)}{2} = \frac{5 \times 4}{2} = 10
\]

Thus, the correct answer is:

**b. 10**

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. What is the handshake formula for 6 people?
2. How does the combination formula apply to real-world problems?
3. How many handshakes occur if 8 people are involved?
4. How does the number of handshakes grow as the number of people increases?
5. Can the handshake problem be visualized using a graph?

**Tip:** The combination formula \( \binom{n}{2} \) is useful for counting pairwise interactions in many scenarios, not just handshakes.

Arkady, Boris, Vladimir, Grigory and Dmitry shook hands when they met (each shook hands with each other once). How many handshakes were done?

a. 5
b. 10
c. 15
d. 25

Learn how to calculate the total number of handshakes when 5 people meet and each shakes hands once. Using the combination formula C(n, 2) = n(n-1)/2, this problem teaches the basics of combinatorics and counting. Suitable for students in Grades 8-10.

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