Week 8-9: The Inclusion-Exclusion Principle
March 31, 2005
1
The Inclusion-Exclusion Principle
Let S be a nite set, and let A, B, C be subsets of S. Then
|A B| = |A| + |B| |A B|,
|A B C| = |A| + |B| +
The Pigeonhole Principle
1
Pigeonhole Principle: Simple form
Theorem 1.1. If n + 1 objects are put into n boxes, then at least one box contains two or more objects.
Proof. Trivial.
Example 1.1. Among
Permutations and Combinations
March 11, 2005
1
Two Counting Principles
Addition Principle. Let S1 , S2 , . . . , Sm be subsets of a nite set S. If S = S1 S2 Sm , then
|S| = |S1 | + |S2 | + + |Sm |.
Mu
Homework 3
Chapter 5, pp.153: 11, 12, 22, 31, 40, 45.
Chapter 6, p.185: 4, 11, 16, 24, 30.
1. Use combinatorial reasoning to prove the identity
n
k
n3
k
n1
k1
=
n2
k1
+
+
n3
k1
.
Proof. Let S be a set
Week 10-11: Recurrence Relations and Generating Functions
April 20, 2005
1
Some Number Sequences
An innite sequence (or just a sequence for short) is an ordered array
a0 , a1 , a2 , . . . , an , . . .
Week 12-13: Special Counting Sequences
May 5, 2005
We have considered several special counting sequences. For instance, the sequence n! counts the number of
permutations of an n-set; the sequence Dn c
What is combinatorics?
1
What is combinatorics?
Examples of combinatorial problems:
(1) Finding the number of games that n teams would play if each team played with every other team
exactly once.
(2)
Week 6-7: The Binomial Coecients
March 16, 2005
1
The Pascal Formula
Theorem 1.1 (Pascals Formula). For integers n and k such that 1 k n,
n
k
The numbers
n
2
=
n(n1)
2
=
n1
k
n1
k1
+
.
(n 2) are trian
Homework 1
Chapter 1, p.21: 2, 5, 8, 13, 30, 33, 35
Chapter 2, p.41: 5, 9, 14, 15, 17
Supplementary Exercises
1. For the game of Nim, let us restrict that each player can move one or two coins. Find t
Generating Permutations and Combinations
March 10, 2005
1
Generating Permutations
We have learned that there are n! permutations of cfw_1, 2, . . . , n. It is important in many instances to generate a