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

Learn how to use generating function models to solve combinatorial problems using an algebra-calculus approach. Explore examples and applications.

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

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

  2. 6.1. Generating Function Models • Algebra-Calculus approach. • We are given a finite or infinite sequence of numbers a0, a1, ..., an, ... • Then the generating function g(x) for an is given by: • g(x) = a0 + a1x + ... + a2xn + ...

  3. 6.1. Generating Function Models • Homework (MATH 310#8F): • Read 6.2. • Turn in 6.1: 6,8,10,22 • Volunteers: • ____________ • ____________ • Problem: 22.

  4. Combinatorial Approach • (a + x)(a + x)(a + x) = aaa + aax + axa + xaa + axx + xax + xxa + xxx. • What is the coefficient at x2? • axx + xax + xxa a 3x2. [3 = C(3,2)] • (1 + x)(1 + x)(1 + x) = 111 + 11x + 1x1 + x11 +1xx + x1x + xx1 + xxx = 1 + 3x + 3x2 + x3 • In general, in (1 + x)n the coefficient at xr is C(n,r).

  5. Question • What is the meaning of the coefficient at x5 in (1 + x + x2)4? • xxxx2 + xxx2x + xx2xx + x2xxx + 1xx2x2 + 1x2xx2 + 1x2x2x + .... ( 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.

  6. Example 1 • Find the generating function for ar, 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 + x2 + x3 )4

  7. 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 + x2 + x3 + x4)3 • (b) (1 + x + x2 + x3 + ... )3

  8. Example 3 • Find the generating function for ar, 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 + x2 + x4 + x6 + x8 + x10)2 (x3 + x4 + x5)3

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