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

Question

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?

Adrian Alexander · Accepted Answer

Shaking hands in a group involves pairings of two people in all possible ways

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#

Nora Amin · Answer

see a solution process below;

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#

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?

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?