Combinatorics. Combinatorics began as a formalized treatment of efficient ways of counting certain collections of objects which arise relatively often. Nowadays the word ‘combinatorics’ can be used to refer to pretty much all of finite mathematics, and the original field is more specifically called “enumerative combinatorics”.
The things in parentheses are not fractions (they don’t have a fraction bar). These are called binomial coefficients and (n [over] k) represents the number of ways to choose k balls from a set of n balls. Pascal’s triangle is an arrangement of these numbers where (n [over] k) is the k-th number in the n-th row; it is very famous to be a useful way of visualizing many of the properties of the binomial coefficients.
I’m experimenting a little with how to display the theorem statements. Not really sure what I like yet.