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

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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?

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#

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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#

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