How many handshakes are possible in a room with 30 people?

Answer 1

in endless time, endless handshakes, if you don't count the same couple twice or more.

but you're appealing to the handshake principle

n(n-1)/2 = 30(29)/2 = 15(29) = 435 handshakes

very similar formula to Euler's n(n+1)/2 for adding up sums of numbers.