C(n,k) = n! / (k! (n-k)!).
Symmetry: C(n,k) = C(n, n-k), which often halves the arithmetic - C(20,18) is C(20,2) = 190.
Pascal's rule: C(n,k) = C(n-1,k-1) + C(n-1,k), from conditioning on whether a particular item is chosen.
The trap. Order. If the selection is ordered, you want permutations, n!/(n-k)!, which is larger by a factor of k!. Deciding "does order matter here?" before computing anything prevents most counting errors.
Worth memorising the small values: C(4,2)=6, C(5,2)=10, C(6,3)=20, C(52,2)=1326.