Question

there are 12 people at a party. each person shakes hands with each of th other guests. how many hadshakes will there be in all

Answer 1

each of the 12 shakes hands with 11 others, but A shaking with B is the same as B shaking with A, so
12 x 11/2 = 66

In combination notation:
C(12,2) = 12!/(10!2!) = 66

Answer 2

Another viewpoint:

A shakes the hand of 11 people.
B shakes the hand of 10 people, already having shaken the hand of A.
C shakes the hand of 9 people, already having shaken the hand of A and B.

Therefore, the total number of hand shakes is simply the sum of the numbers from 1 through 11 or, S = n(n + 1)/2 = 11(12)/2 = 66.

Answer 3

1,993,373

Answer 4

You might want to recheck the math. If you start with #12 that's 11 shakes then 10, 9,8,7,6,5,4,3,2. when you get to #2 he shakes with #1 that is 65 #1 has shook with everybody so doesn't need to shake with anyone.