First Way:
Since there are 10 people, and each handshake requires 2 people, this is the same as choosing 2 people out of 10 people.
Thus, the answer will be 10 C 2 = 45 handshakes
Second Way:
The first person will shake hands with 9 people, the second person will shake hands with the remainder 8, the third person with the remainder 7....
Thus, we have 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 handshakes