If there are two people at a party, they can shake hands once. There
is no one else left to shake hands with, so there is only one
handshake total.

 2 people, 1 handshake

If there are three people at a party, the first person can shake hands 
with the two other people (two handshakes).  Person two has already 
shaken hands with person one, but he can still shake hands with person 
three (one handshake).  Person three has shaken hands with both of 
them, so the handshakes are finished.  2 + 1 = 3.

 3 people, 3 handshakes
If there are four people at a party, person one can shake hands with 
three people, person two can shake hands with two new people, and 
person three can shake hands with one person.  3 + 2 + 1 = 6.

 4 people, 6 handshakes

Are you seeing a pattern?

If you have five people, person five shakes four other hands, person 
four shakes three other hands, person three shakes two other hands, 
and person two shakes one hand.  Another way to see it is,

 Person 5  Person 4  Person 3  Person 2

    4    +    3    +    2    +    1    =   10 handshakes total  


====================================================
People at Party               Number of Handshakes

     2                                           1
     3                      1 + 2              = 3
     4                      1 + 2 + 3          = 6
     5                      1 + 2 + 3 + 4      = 10
     6                      1 + 2 + 3 + 4 + 5  = 15
     .
     .
     .                                             n(n-1)
     n                      1 + 2 + ...+ (n-1) =  --------
                                                      2
8(8-1)/2= 28
Each man shakes hands with everyone except his own wife: 28-4=24
No women shake hands with each other: 24 -((4x3):2)= 18