Hand Shake

Direct formula --> [n(n-1)]/2=total handshake where n gives total no. of persons present at party.
Indirect approach ------> let total no of people at party be n.
1st person makes n-1 handshakes.-->(n-1)
2nd person makes n-2 handshakes.-->(n-2)
3rd person makes n-3 handshakes.-->(n-3) ...... so on.......
(n-1)th person makes 1 handshakes.-->(1)
1st person makes 0 handshakes.-->0
Total handshake=(n-1)+(n-2)+(n-3)+ .......... +1+0=sum of natural no.s up to n-1
total handshake=(n-1)(n)/2=66 (given) solving this we have n=12

using combination formula nC2=66

solving n=12

If nos of people is n

C(n,2) = 66

=> n!/{(n-2)!*2!}

=> n(n-1)(n-2)!/{(n-2)!*(2x1)} = 66

=> n^2-n = 132

=> n^2-n-132 = 0

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

Either n-12 = 0 => n = 12

or n+11 = 0 => n = -11 (n can't be -ve)

So nos of people is 12