At the end of a banquet 10 people shake hands with each other. how many handshakes will there be in total?

10 people, who obviously wouldn't shake hands with people twice, or shake hands with themselves.
Let's just label them, P meaning person, the numbers after being who they shake hands with:
P1: 2, 3, 4, 5, 6, 7, 8, 9, 10 = 9 handshakes
P2: 3, 4, 5, 6, 7, 8, 9, 10 = 8 handshakes
P3: 4, 5, 6, 7, 8, 9, 10 = 7 handshakes
P4: you get the picture, this guy shakes 6 hands, etc
P5:5 handshakes
P6:4 handshakes
P7:3 handshakes
P8:2 handshakes
P9:1 handshake
P10: everyone has already shaken hands with him by now!
Now add all the handshakes together:
45 handshakes, total