DEAN: So, how do you know how to do this? [tie a bow-tie]
CHRISTOPHER: Seventeen cotillions, a dozen debutante balls, and a brief but scarring experiment with the Children of the American Revolution.
In the US, cotillions are the classes given in dancing and etiquette to prepare girls and boys for society. A cotillion ball is given at the end, which is not only a celebration, but also a preparation for the debutante ball which will come later. This ball itself is often known as a cotillion. I’m not sure, but I think Christopher means that he attended seventeen of such balls while he was growing up, as well as twelve debutante balls.
The National Society Children of the American Revolution, founded in 1895, is a youth organisation for those under the age of 22 who are descended from someone who served in the American Revolution, or gave material aid to its cause. The Daughters of the American Revolution is thus one of its parent organisations.