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How many people in party?

(28 votes, average: 3.61 out of 5)

November 20, 2012 by puzzler 6 Comments

At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party?
See Solution: How many people in party?Hide Solution

lets say there are n persons
first person shakes hand with everyone else: n-1 times(n-1 persons)
second person shakes hand with everyone else(not with 1st as its already done): n-2 times
3rd person shakes hands with remaining persons: n-3

So total handshakes will be = (n-1) + (n-2) + (n-3) +…… 0;
= (n-1)*(n-1+1)/2 = (n-1)*n/2 = 66
= n^2 -n = 132
=(n-12)(n+11) = 0;
= n = 12 OR n =-11
-11 is ruled out so the answer is 12 persons.

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Comments

  1. Arjun says

    January 15, 2022 at 11:34 pm

    For Basic understanding, split the problem in small number first.
    Lets assume there are 3 people. ABC. So from these three we get 6 combinations(AB,AC,BA,BC,CA,CB), 6 can be written as 3*2=6,means n(n-1).
    but in the above combinations, there is a double counting(AB, BA) is same. so we have to divide by 2, which results in n(n-1)/2. this is also a combination formula ncr which n!/r(n-r)!, r is considered as 2 because only two ppl can shake.

    Reply
  2. Margaret says

    November 23, 2021 at 5:14 pm

    How did you get the 4 vertices

    Reply
  3. Kamal dev rana says

    January 26, 2021 at 3:16 am

    how n(n-1) = 132 , results in 12

    Reply
  4. The_Geek says

    October 26, 2014 at 5:04 pm

    we can think this problem as a complete graph problem.
    In a 4 vertices complete graph, no. of edges are n(n-1)/2, i.e. 6
    Similarly here n(n-1)/2=66
    so n(n-1)=132,
    hence n=12.

    Reply
  5. shubham says

    September 25, 2014 at 8:48 pm

    Ans is 12 by te formula of e=mc^2 lol:p

    Reply
  6. vinay says

    December 28, 2012 at 12:56 pm

    12 people in the party
    (n)c(2) = 66.

    Reply

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