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?

Question
Answer 1

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#

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

see a solution process below;

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

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

#n(n - 1)#

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

#(n(n – 1))/2#

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.

#380/2 = 190#

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

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

#n = "total number of persons"#

#r = "number of handsakes"#

#n = 20#

#r = 2#

#(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#

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7