Question

20 people shake hands with each other. How many handshakes will be there in total?

Solution

Shaking hands in a group involves pairings of two people in all possible waysExplanation:Say we have $$N$$ people in the room. So to shake hands we have to pair each one of these $$N$$ with each one of the rest of people in the room. So we have  $$N \cdot (N - 1)$$ possible pairings.However, in this number we have actually counted each pairing twice; when, say $$person-1$$ shakes hands with $$person-2$$, and when $$person-2$$ shakes hands with person1. It is only one handshake.Thus, the correct number is half of that $$\dfrac{N \cdot (N - 1)}{2}$$$$=\dfrac{20(20-1)}{2}=10\times19=190$$Mathematics

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