For each integer \(n \ge 0\) and integer \(k\) with \(0 \le k \le n\) there is a number
read “\(n\) choose \(k\text{.}\)” We have:
\({n\choose k} = |\B^n_k|\text{,}\) the number of \(n\)-bit strings of weight \(k\text{.}\)
\({n \choose k}\) is the number of subsets of a set of size \(n\) each with cardinality \(k\text{.}\)
\({n \choose k}\) is the number of lattice paths of length \(n\) containing \(k\) steps to the right.
\({n \choose k}\) is the coefficient of \(x^ky^{n-k}\) in the expansion of \((x+y)^n\text{.}\)
\({n \choose k}\) is the number of ways to select \(k\) objects from a total of \(n\) objects.