Six business people meet for lunch and shake hands with each other. How many handshakes are there?
Solution
If two people shake hands there is one handshake.
If a third person comes and shakes hands with the other two people that makes a total of 1 + 2 = 3 handshakes.
If a fourth person comes and shakes hands with the other three people that makes a total of 3 + 3 = 6 handshakes.
If a fifth person comes and shakes hands with the other four people that makes a total of 6 + 4 = 10 handshakes.
If a sixth person comes and shakes hands with the other five people that makes a total of 10 + 5 = 15 handshakes.
Another way of looking at the problem is that each person has to shake hands with all the others. If there are 6 people each person has 5 handshakes to make. But each time a handshake occurs there are 2 people involved. This means that you only need ½(6 x 5 ) = 15.
Extension
How many handshakes are there at the meeting if people come in pairs and shake hands with everyone except their own partners.