Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 2
1. Case . Fill in the blanks:
# choices :
5
5
5
5
5
4
3
2
5
5
5
2
2
3
4
2
The answer is 54 .
Case cfw_a. Fill in the blanks left to right:
#
MATH413
HW 11 - bonus questions
1:
Does there exist a BIBD with parameters b = 10, v = 8, r = 5, and
k = 4?
2:
Does there exist a BIBD with parameters b = 20, v = 18, r = 10, and
k = 9?
3:
Construct three mutually orthogonal latin squares of order four.
4
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 3
1. For 1 k 22 we show that there exists a succession of consecutive days during which
the grandmaster plays exactly k games. For 1 i 77 let
MATH413
HW 10
due Apr 25 strictly before class
Solutions without explanation will receive no points.
1: P.320, # 22(a) Compute p6 (the partition number of 6) and construct a
Hasse diagram of partialy ordered set P6 where Pn contains all partitions of
n (i
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 6
1. Dene the set S = cfw_1, 2, . . . , 104 . Let A (resp. B) (resp. C) denote the set of integers in
S that are divisible by 4 (resp. 5) (res
MATH413
HW 6
due Mar 14 before class
1: (P. 199, #13) Determine the number of permutations of cfw_1,2,. . . ,9 in
which at least one odd integer is in its natural position.
Solution:
For i = 1, 3, 5, 7, 9, let Ai denote the set of permutations of cfw_1, .
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 8
1. See the solution to Problem 41 in Chapter 7.
2. Let Pn denote the set of permutations of the multiset cfw_n 1, n 1. Let Mn denote the
set
MATH413
HW 1
due Feb 1 before class
1:
How many ways are there to pick a man and a woman who are not
husband and wife from a group of n married couples?
Solution:
Total number of possibilities to pick man and woman is n2 . Number of bad
choices is n. Henc
MATH413
HW 3
due Feb 19 before class, answer without justication will receive 0 points.
1: (P. 66, #50) In how many was can ve identical rooks be placed on
the squares of an 8-by-8 board so that four of them form the corners of a
rectangle with sides para
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 5
1. For an integer k and a real number n, we show
n
k
=
n1
n1
+
.
k1
k
First assume k 1. Then each side equals 0. Next assume k = 0. Then eac
Spring 2016
Introduction to Discrete Mathematics
Practice for Final
May 4, 2016.
Name:
UIN:
Instructions
Write your name and UIN on top of this
page.
Please put your I-card face-up on your
desk.
Make sure you have the total of 2 pages,
numbered from 1
Spring 2016
Introduction to Combinatorics
Exam 3
April 25, 2016.
Name:
UIN:
Instructions
Write your name and UIN on top of this
page.
Please put your I-card face-up on your desk.
Make sure you have the total of 7 pages,
numbered from 1 to 7.
You are n
Spring 2016
Introduction to Discrete Mathematics
Practice for Midterm 1
February 10, 2016.
Name:
UIN:
Instructions
Write your name and UIN on top of this
page.
Please put your I-card face-up on your
desk.
Make sure you have the total of 6 pages,
number
Spring 2016
Introduction to Discrete Mathematics
Solutions of Practice for Final
May 9, 2016.
Name:
UIN:
Instructions
Write your name and UIN on top of this
page.
You can use the back of the test paper as
scratch paper. If you need more, ask the
instruc
Math 413 HW5 Due: Oct 7th
1. (Q11) Use combinatorial reasoning to prove the identity:
= n1 + n2 +( n3)
(nk)(n3
k ) (k 1) (k 1) k 1
Let S be a set of n elements. We distinguish 3 of the elements of S and
denote it by a, b and c. We want to count the number
Page 2 of 6
MATH 413
Name:
(1) Determine the number of integral solutions to
x1 + x2 + x3 + x4 + x5 = 2015
subject to
x1
100, x2
30, x3
20, x4
5, x5
10.
MATH 413
Page 3 of 6
(2) Show that any set of 76 positive integers
100 must contain 4 consecutive inte
Math 413 HW7 Due: Oct 21st
1. (Q17) Determine the number of permutations of the multiset
S=cfw_3*a, 4*b, 2*c
Where for each type of letter, the letters of the same type do not
appear consecutively.
Let X denote the permutation of S. Let A, B and C denote
Math 413 HW6 Due: Oct 14th
1. (Q30) Find and prove a formula for:
r ,s , t 0
r +s+t =n
( mr )(ms )( mt )
1
2
3
Where the summation extends over all nonnegative integers r, s,
and t with sum r+s+t=n.
The hint suggests that the
r ,s , t 0
r +s+t =n
( mr )(m
Math 413 HW2 Due: Sep 14th
1. (Q33) Determine the number of 10-permutations of the multiset
S=cfw_3.a, 4.b, 5.c.
It is clear that in order to get a 10-permutations there need to be exactly
2 letters missing. There are 6 different way to do this, so we lef
Math 413 HW1 Due: Aug 31th
1. How many nonempty words can be formed from three As and five
B's? (The "words" are "mathematical words" not necessarily
"dictionary words". For example B, BAAB, ABABABBB are possible
words you need to count. )
With 3 A and 5
Math 413 HW2 Due: Sep 7th
1. In how many ways can 2 red and 4 blue rooks be placed on a 8 by
8 board so that no two rooks can attack on another?
We can start with picking 6 rows for the rocks to be putted in, there are
C(8, 6) ways to do this. Then we pic
U. of Illinois MATH 413 Test 2 Fall 2016
Answer as many problems as you can. Each question is worth 6 points
(total points is 30). Show your work. An answer with no explanation will receive no credit. Write your name on the top right
corner of each page.
U. of Illinois MATH 413 Test 2 Fall 2016
Answer as many problems as you can. Each question is worth 6 points
(total points is 30). Show your work. An answer with no explanation will receive no credit. Write your name on the top right
corner of each page.
University of Illinois MATH 413 Test 1 Fall 2016
Answer as many problems as you can. Each question is worth 6 points
(total points is 30). Show your work. An answer with no explanation
will receive no credit. Write your name on the top right corner of eac
U. of Illinois MATH 413 Test 1 (E13) Fall 2016
Answer as many problems as you can. Each question is worth 6 points
(total points is 30). Show your work. An answer with no explanation
will receive no credit. Write your name on the top right corner of each
Spring 2016
Introduction to Discrete Mathematics
Practice for Midterm 1
February 10, 2016.
Name:
UIN:
Instructions
Write your name and UIN on top of this
page.
Please put your I-card face-up on your
desk.
Make sure you have the total of 6 pages,
number
MATH413
MIDTERM 2
March 16 10:00-10:50am
Name: .
Answer as many problems as you can. Show your work. An answer with
no explanation will receive no credit. Write your name on the top right corner
of each page.
Problem 1 Problem 2 Problem 3 Problem 4 Proble
MATH413
HW 3
due Feb 15 before class
1: (P. 63, #28) A secretary works in a building located nine blocks east
and eight block north of his home. Every day he walks 17 blocks to work.
(See the map that follows.)
(a) How many different routes are possible f
MATH413
HW 8
due Apr 13 strictly before class
1: P.260, #23 Let be a real number. Let the sequence h0 , h1 , h2 , . . . , hn , . . .
be defined by h0 = 1, and hn = ( 1) ( n + 1), (n 1). Determine
the exponential generating function for the sequence.
2: So