The most special class that emerges in counting problems, or enumerations as they are called, is that of the binomial coefcients,so named as they arise as the multipliers of powers of x when the binomial expression(1+x)n is expanded. The binomial coefcient C(n，r)is the number of different ways we may construct a set of size r from one of size n. For example, C(4，2)=6, as there are six pairs(taken without regard to order within a pair)that can be chosen from a group of four:for example, if we have four children,Alex, Barbara, Caroline, and David, there are six ways that we can select a pair from this group:AB, AC, AD, BC, BD, and CD.