Question

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

We use this formula for this but why?
N(N-1)/2
20(20-1)/2
20 * 19/2
10 * 19
190
Can anybody explain this formula?

Answer 1

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*(N-1)` possible pairings.

However, in this number we have actually counted each pairing twice; when, say `person1` shakes hands with `person2`, and when `person2` shakes hands with `person1`. It is only one handshake.

Thus, the correct number is half of that `(N* (N-1))/2`

Answer 2

see a solution process below;

Explanation:

First using your method which is the formula for handshakes;

Note that;

`n = "total number of people that will shake hands"`

but since all (total) can't shake hands with themselves, hence we subtract one individual to start the handshaking..

`(n - 1) = "people would each shake hands"`

hence we multiply both total numbe persons with people that will shake hands;

`n(n - 1)`

but that counts every handshake twice, so we have to divide by `2.`

Therefore;

`(n(n – 1))/2`

Secondly;

You know that the total number of persons is `20`, so every person shakes hands with `19` persons..

It then mean that, there are `20×19=380` handshakes.

But by every handshake two persons are involved.

Hence;

`380/2 = 190`

Therefore, `380` is the result of double-counting, which gives `190` handshakes.

Thirdly;

I just used normal simple combination formula..

Recall; `rArr ^nC_r = (n!)/((n- r)!r!)`

`n = "total number of persons"`

`r = "number of handsakes"`

`n = 20`

`r = 2`

Inputing in the formula we should have;

`(n!)/((n- r)!r!)`

`(20!)/((20 - 2)!2!)`

`(20!)/(18!2!)`

`(20 xx 19 xx 18!)/(18!2!)`

`(20 xx 19 xx cancel(18!))/(cancel(18!)2!)`

`(380)/(2 xx 1)`

`380/2`

`190`