"5 choose 2":



Canxcel the two 2's into the 4

 

Cancel the 3's



You are left with  or  ahndshakings.

If the people are A,B,C,D, and E, they are
 
 1. A shakes hands with B
 2. A shakes hands with C
 3. A shakes hands with D
 4. A shakes hands with E
 5. B shakes hands with C
 6. B shakes hands with D
 7. B shakes hands with E
 8. C shakes hands with D
 9. C shakes hands with E
10. D shakes hands with E

Edwin