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
