Counting pairs
The question
19 offices are connected so that each one can call each of the others directly, and a call from A to B is not the same as a call from B to A.
How many different calls are possible?
Count the pairs without listing them
how many:19
19−1=18
19×18=342
342
- There are 19 of them, and every one of them reaches every other. There is no need to write the pairs out.
- Take one of the 19 and count who they meet. Not themselves, so 19 − 1 = 18 others — and the same is true for every one of them.
- So all together that is 19 × 18 = 342.
- Here A to B and B to A count as two different things, so there is no halving. The answer is 342.
How it works.
- 01
There are 19 of them, and every one of them reaches every other. There is no need to write the pairs out.
- 02
Take one of the 19 and count who they meet. Not themselves, so 19 − 1 = 18 others — and the same is true for every one of them.
- 03
So all together that is 19 × 18 = 342.
- 04
Here A to B and B to A count as two different things, so there is no halving. The answer is 342.