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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 a0, a1, ..., an,
...
 Then the generating function g(x)
for an is given by:
 g(x) = a0 + a1x + ... + a2xn + ...
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6.1. Generating Function
Models
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Homework (MATH 310#8F):
• Read 6.2.
• Turn in 6.1: 6,8,10,22
• Volunteers:
• ____________
• ____________
• Problem: 22.
Combinatorial Approach
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(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).
Question
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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 typeNumber of distributions of 5 identical objects
into four boxes with at most 2 objects in any
box.
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
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Example 2
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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
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 + x + x + x + x + x ) (x + x + x )
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2
4
6
8
10 2
3
4
5 3