Hand Shake problem

There are 20 people in a room every person must shake hands with every other person in the room, How many hand shakes have taken place?

1. First you would have shaken every other persons hand.
2. then you would count them.
3. in the end you would have shaken 190 people's hands.
4. 19+18+17+16+15+14+13+12+11+10+9+8+7+6+5+4+3+2+1+0= 190 =]
5. (n-1)+( n-2)+ (n-3)+(n-4)+(n-5)+(n-6)+(n-7)+(n-8)+(n-9)+(n-10)+(n-11)+(n-12)+(n-13)+(n-14)+(n-15)+(n-16)+(n-17)+(n-18)+(n-19)