98

8

Sets and Combinatorics

where aa and bb can also be compound expressions, can be derived by combinatorial

reasoning; for example, left parenthesis a plus b right parenthesis Superscript 5 Baseline equals left parenthesis a plus b right parenthesis left parenthesis a plus b right parenthesis left parenthesis a plus b right parenthesis left parenthesis a plus b right parenthesis left parenthesis a plus b right parenthesis(a + b)⁵ = (a + b)(a + b)(a + b)(a + b)(a + b), and to

generate the terms, an aa or bb is chosen from each of the five factors.

Problem. Generalize the binomial theorem by replacing the binomial a plus ba + b by a

multinomial a 1 plus a 2 plus midline horizontal ellipsis plus a Subscript ra1 + a2 + · · · + ar.