Math 220
Lecture 9
Baili Min
July 24, 2009
1
Combinatorics
Sorry I wont give details about solutions to those problems. You are responsible to take notes in
class. Dont hate me, its for your sake.
Review Problems
Have a quick review of last lecture, and d
Math 220
Lecture 13
Baili Min
July 30, 2009
1
Graph Theory
Connectivity
Suppose employees in a company have some telephone lines so that they can contact some other
employees. The graph is shown below.
Now the company wants to cut the cost so it wants som
Math 220
Lecture 10
Baili Min
July 27, 2009
1
Combinatorics
Another Example
Draw a planar grd that is 30 squares wide and 15 squares high. How many different nontrivial
rectangles, that is, rectangles with positive width and height, can be drawn by using
Math 220
Lecture 8
Baili Min
July 23, 2009
1
Combinatorics
Permutation with Multiplicities
CASE 1: No Restriction
Problems
1. Suppose we have a box containing 100 different balls labeled from 1 through 100. Everytime
we take out a ball, record its number
Math 220
Lecture 7
Baili Min
July 22, 2009
1
Combinatorics
The Binomial Theorem
What do you get if you expand (x + y)2 , (x + y)3 and (x + y)4 ?
How about (x + y)9 ? (x + y)15 ? (x + y)200 ? Do you really want to perform the multiplication
for 200 times?
Math 220
Lecture 6
Baili Min
July 20, 2009
1
Combinatorics
Pigeonhole Principle
Before we start serious materials, we begin with some questions:
Question 1:
There are 100 letters to be delivered to 99 mailboxes. What will happen?
Question 2:
There is a da
Math 220
Lecture 5
Baili Min
July 17, 2009
1
Number Theory
Public-Key Cryptography
Idea
As we have seen, we need to get some pattern which is sort of one-way, that is: everyone knows
that pattern to encrypt but only the receiver knows how to decrypt. In m
Math 220
Lecture 4
Baili Min
July 16, 2009
1
Number Theory
Question: Now you are in this class and get so bored and want to tell your friend a few rows
behind you that the teacher is SO BAD. You decide to send him/her a message note (because you
happen to
Math 220
Lecture 3
Baili Min
July 15, 2009
1
Number Theory
In a third-centry A. D. Chinese book Sun Tzus Calculation Classic one problem is recorded
which can be translated into English as:
Suppose we have an unknown number of objects. When counted in thr
Math 220
Lecture 2
Baili Min
July 14, 2009
1
Number Theory
Suppose p N and p > 1. If the only positive factors of p are 1 and itself, then we say p is a prime
number. Otherwise, we call it a composite number.
Question: Can you give examples of prime numbe
Math 220 Homework 4
Solutions
Solve the following problems:
1. How many words of length 9 can be formed by 26 English letters?
Solution: Permutation with multiplicities, no restriction: 269 .
2. How many words of length 9, each of which has 3 f s, 3 ts an
Math 220 Homework 2
Baili MIN
Solutions
Prove the followings:
(1) Suppose n is a non-zero natural number. Then (an , bn ) = (a, b)n .
Please refer to your notes taken in class.
(2) Suppose n > 1 is an integer. If for any integer m either n|m or (n, m) = 1
Math 220
Lecture 1
Baili Min
July 13, 2009
1
Number Theory
Numbers like 1, 2, 0, -1, -99, . . . are called integers. The set of all integers in denoted as Z.
An interesting subset of Z is the collection of all those positive numbers in Z, denoted as N.
Fo
Math 220 Homework 1
Baili MIN
Solutions
Divisibility
Prove the following statements:
Suppose a, b, c are integers.
(i) If b|a and c = 0, then bc|ac. The converse is also true. In particular, b|a is equivalent to
()b|()a.
b|a if and only if a = bd for some
Math 220 Homework 3
Solutions
Solve the following problems:
1. There are three identical balls and four different boxes: white, black green and brown. Now
put those three balls to those four boxes such that each box will contain at most one ball. Then
how