Question

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

Answer 1

let the total no, of persons who attended the party b n

1st person shakes hand with everyone else: n-1 times(obviously he will not shake hand with himself.)

2nd 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 times

So total handshakes will be = (n-1) + (n-2) + (n-3) +…… 0;

= (n-1)*(n-1+1)/2 = (n-1)*n/2 = 66 (using sum of n consecutive natural numbers formula)

= n^2 -n = 132

=(n-12)(n+11) = 0;

= n = 12 OR n =-11

-11 is ruled out so the answer is 12 persons.
there were 12 persons in the party.

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