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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 ways

Explanation:

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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