Binomial coefficient

The binomial coefficient is the same number \(C(n,k)\) from combinations with a different notation, read «\(n\) choose \(k\)», which stresses that it is a number in its own right. For example, \(\dbinom{6}{2}=\dfrac{6!}{2!\,4!}=15\): there are 15 possible pairs in a group of 6 people. In practice it is computed by cancelling factors, \(\dbinom{n}{k}=\dfrac{n(n-1)\cdots(n-k+1)}{k!}\), without evaluating full factorials.

It satisfies the symmetry \(\dbinom{n}{k}=\dbinom{n}{n-k}\), the extreme values \(\dbinom{n}{0}=\dbinom{n}{n}=1\) and Pascal's rule \(\dbinom{n}{k}+\dbinom{n}{k+1}=\dbinom{n+1}{k+1}\). The latter is explained by counting: the subsets of \(k+1\) elements of a set of \(n+1\) either avoid a fixed element or contain it. It builds Pascal's triangle, where each number is the sum of the two above it.

Adding up a whole row counts all subsets of an \(n\)-element set, so \(\sum_{k=0}^{n}\dbinom{n}{k}=2^n\). The coefficients also count grid paths: going from one corner to the opposite one with \(a\) steps right and \(b\) steps up gives \(\dbinom{a+b}{a}\) paths, since it is enough to decide which of the \(a+b\) steps go right.

Typical mistake: confusing \(\dbinom{n}{k}\) with a fraction or with \(n^k\); it is an integer that grows quickly toward the middle of the row and is symmetric. These same numbers are the coefficients in the binomial theorem.