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| + |C| |A B| |A C| |B C| + |A B C|.
Let P1 , P2 , . . . ,
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 13 people there are two who have their birthdays in the
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 |.
Multiplication Principle. Let S1 , S2 , . . . , Sm be nit
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 of n elements, and let a, b, c be distinct elements of
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 , . . .
of countably many real or complex numbers, and is usua
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 counts the number of derangements of an n-set; and the t
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) Constructing a magic square.
(3) Attempting to trace th
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 triangle numbers, i.e.,
The numbers n = n(n1)(n2) (n 3) are
Homework 2
Chapter 3, pp.76: 9, 14, 20, 23, 27, 36.
Chapter 4, p.116: 4, 6, 7, 8, 9, 11, 12, 23, 28.
Supplementary Exercises
1. Find the number of ways to select m numbers from cfw_1, 2, . . . , n so that no two numbers are consecutive.
2. A move of a per
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 the winning strategy
for each player.
2. Let n be a posi
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
list of such permutations. For example, for the permut