discrete math problem college level question

[afang]

Suppose 2n people sit on a round table and are shaking hands in
pairs. Suppose that etiquette is observed and no 2 shakes cross. Let
S_n be the number of possible shaking hands arrangements of this sort.

Determine S_10.

[phoenixthoth]

I recommend trying to work this out for n=2, 3, and 4 and then find a pattern.

You'll need to find a formula for the sum of the first k whole numbers which is probably in your book somewhere.