# 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?

Mar 26, 2018

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 $r e s t$ of people in the room. So we have $N \cdot \left(N - 1\right)$ possible pairings.

However, in this number we have actually counted each pairing twice; when, say $p e r s o n 1$ shakes hands with $p e r s o n 2$, and when $p e r s o n 2$ shakes hands with $p e r s o n 1$. It is only one handshake.

Thus, the correct number is half of that $\frac{N \cdot \left(N - 1\right)}{2}$

Mar 26, 2018

see a solution process below;

#### Explanation:

First using your method which is the formula for handshakes;

Note that;

$n = \text{total number of people that will shake hands}$

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

$\left(n - 1\right) = \text{people would each shake hands}$

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

$n \left(n - 1\right)$

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;

$\frac{380}{2} = 190$

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!)

$n = \text{total number of persons}$

$r = \text{number of handsakes}$

$n = 20$

$r = 2$

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!)

$\frac{380}{2 \times 1}$

$\frac{380}{2}$

$190$