I get a little different result from tertre
Note that the
10th person shakes 9 hands
9th person shakes 8 hands
8th person shakes 7 hands
.....
2nd person shakes 1 hand
So.....the number of handshakes seems just to be the sum of the first 9 positive integers =
9 * 10 / 2 = 45 handshakes