Handshakes

A. There are 3 people in a room. Assuming each person is equally polite and shakes hands with every other person only once, how many handshakes occur? (Note: you obviously don't shake hands with yourself.)

B. There are 16 people in a room. Assuming each person is equally polite and shakes hands with every other person only once, how many handshakes occur? (Note: you obviously don't shake hands with yourself.)

C. There are n people in a room. Assuming each person is equally polite and shakes hands with every other person only once, how many handshakes occur? (Note: you obviously don't shake hands with yourself.)

My answer to A, was 3. The 1^st shakes the 2^nd and 3^rd person which is 2 handshakes. The 2^nd person already shook hands with the 1^st person so he shakes hands with the 3^rd person. The 3^rd person already shook hands with everyone so the answer is 3.

My answer to B, is 120. The 1^st person shakes hands with everyone but himself. The 2^nd
person shakes hands with everybody but the 1^st and himself. The pattern keeps on going, everybody shaking hands with everybody but themselves and the people before them. So the answer is
15 (made by the 1^st person) + 14 + 13 + 12 + 11+ 10+9+8+7+6+5+4+3+2+1+0 (the 16^th
guy shook hands with everybody allready.) so the answer is 120.

My answer to C, is [n(n-1)]/2, Everybody shakes hands with everybody but themselves, but if you count it like that, you are counting double the times (you are saying, the N^th person shaking hands with the K^th person is different from the K^th person shaking hands with the N^th person. so it is [N(N-1)]/2