MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 25, Friday, October 31

6.1. Generating Function Models Algebra-Calculus approach. We are given a finite or infinite sequence of numbers a 0, a 1,..., a n,... Then the generating function g(x) for a n is given by: g(x) = a 0 + a 1 x a 2 x n +...

6.1. Generating Function Models Homework (MATH 310#8F): Read 6.2. Turn in 6.1: 6,8,10,22 Volunteers: ____________ Problem: 22.

Combinatorial Approach (a + x)(a + x)(a + x) = aaa + aax + axa + xaa + axx + xax + xxa + xxx. What is the coefficient at x 2 ? axx + xax + xxa  3x 2. [3 = C(3,2)] (1 + x)(1 + x)(1 + x) = x + 1x1 + x11 +1xx + x1x + xx1 + xxx = 1 + 3x + 3x 2 + x 3 In general, in (1 + x) n the coefficient at x r is C(n,r).

Question What is the meaning of the coefficient at x 5 in (1 + x + x 2 ) 4 ? xxxx 2 + xxx 2 x + xx 2 xx + x 2 xxx + 1xx 2 x 2 + 1x 2 xx 2 + 1x 2 x 2 x ( 16) Number of solutions to a + b + c + d = 5, 0 · a, b, c, d · 2. Number of selections of 5 objects from four types with at most 2 of each type- Number of distributions of 5 identical objects into four boxes with at most 2 objects in any box.

Example 1 Find the generating function for a r, the number of ways to select r balls from a pile of three green, three white, three blue and three gold balls. Answer: (1 + x + x 2 + x 3 ) 4

Example 2 Use a generating function model for the problem of counting all selections of six objects from three types of objects with repetition of up to four objects of each type. Also model the problem with unlimited repetition. Answer: (a) (1 + x + x 2 + x 3 + x 4 ) 3 (b) (1 + x + x 2 + x ) 3

Example 3 Find the generating function for a r, the number of ways to distribute r identical objects into five distinct boxes with an even number of objects not exceeding 10 in the first two boxes and between three and five in the other boxes. Answer: (1 + x 2 + x 4 + x 6 + x 8 + x 10 ) 2 (x 3 + x 4 + x 5 ) 3