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.