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Counting pairs, step by step

Twelve people, everyone shakes hands with everyone else once.

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The question

How many different calls are possible?

Count the pairs without listing them

how many:19
19−1=18
19×18=342
342
  1. One number to start from: there are 19 of them, and every one of them reaches every other. Writing the pairs out is what this walk is avoiding.
  2. Take one of the 19 and count who they meet. Not themselves, so 19 − 1 = 18 others. That is true of every single one of them, which is what makes this countable without writing anything out.
  3. So all together that is 19 × 18 = 342. Every one of the 19, meeting 18 others.
  4. And here it stops, because this question counts each direction separately — A to B and B to A are two different things. So 342, and no halving.

How it works.

  1. 01

    One number to start from: there are 19 of them, and every one of them reaches every other. Writing the pairs out is what this walk is avoiding.

  2. 02

    Take one of the 19 and count who they meet. Not themselves, so 19 − 1 = 18 others. That is true of every single one of them, which is what makes this countable without writing anything out.

  3. 03

    So all together that is 19 × 18 = 342. Every one of the 19, meeting 18 others.

  4. 04

    And here it stops, because this question counts each direction separately — A to B and B to A are two different things. So 342, and no halving.

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