Imagine that the 5 people form a straight line. The person on the right begins the hugging routine.
He hugs 4 other people. He then goes to the main table and sits down.
The person who is now the new right most person hugs the other 3. then he goes and sits down
The person who is the new right most person hugs the other 2. Then he goes and sits down.
The two remaining pair exchange 1 hug and then they sit down.
The total is 4 + 3 + 2 + 1 = 10.
Question
Is there any other way to do this? Yes. But that is another question.
