Round 1

Put 5 people standing in a line

The other 5 people shake hands with them, that's 5 handshakes.

Round 2

Let the 5 people in the line stay where they are and move the other 5 onto the next person, with the original first of them moving around to the end.

That's another 5 handshakes.

Continue to rotate the 2nd group to get 5+5+5 more handshakes.

In this way each person shakes hands with 5 other people.

However, that's not theonlyway.

In total, there are 5(5)=25 unique handshakes.

However, there are alternative versions of these 25 handshakes.