This is a classic math problem with strong ties to geometry and statistics. Essentially, it asks "How many different handshakes are possible in a room with "n" people?" This can be anaylzed as a pattern beginning with 1 person, 0 handshakes; 2 persons, 1 handshake; and so forth, or, it can be deduced in the following manner:
Suppose there are 20 people in a room. Each of those 20 would shake hands with 19 others. Thus, we have 20 * 19. However, this would count the handshakes between each pair twice. Consequently, we must divide by two, giving us the formula:
20*19/2 or, generalized to n(n-1)/2.
This is also the formula for combinations of n things taken two at a time and for the maximum number of interescetions possible among "n" lines on a plane.