Week 8-9: The Inclusion-Exclusion Principle
March 31, 2005
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
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.
Example 1.1. Among 13 people there are two who have their birthdays in the
Permutations and Combinations
March 11, 2005
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
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
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
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?
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
(2) Constructing a magic square.
(3) Attempting to trace th
Week 6-7: The Binomial Coecients
March 16, 2005
The Pascal Formula
Theorem 1.1 (Pascals Formula). For integers n and k such that 1 k n,
(n 2) are triangle numbers, i.e.,
The numbers n = n(n1)(n2) (n 3) are
Chapter 3, pp.76: 9, 14, 20, 23, 27, 36.
Chapter 4, p.116: 4, 6, 7, 8, 9, 11, 12, 23, 28.
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
Chapter 1, p.21: 2, 5, 8, 13, 30, 33, 35
Chapter 2, p.41: 5, 9, 14, 15, 17
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
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