FINAL EXAM SOLUTIONS
Student Name and ID Number
MATH 3012 Final Exam, December 15, 2010, WTT
Note. The answers given here are more complete and detailed than students are expected to
provide when taking a test. The extra information given here should help
MATH 3012 A
Summer 2012
Homework 6
Due 07/18/2012
1. Can a simple graph have 5 vertices, each with degree 6?
(5)
2. A graph has 21 edges, has 7 vertices of degree 1, three of degree 2, seven of degree 3, and the
rest of degree 4. How many vertices does it
Applied Combinatorics
Preliminary Edition
February 15, 2015
Mitchel T. Keller
Washington & Lee University
William T. Trotter
Georgia Institute of Technology
Copyright c 20082015 Mitchel T. Keller and William T. Trotter.
This text is licensed under the Cre
MATH 3012 Homework Problems and Solutions Chapter 2, Spring 2009, WTT and MTK
Note: These problems are grouped into two categories. The rst set consists of basic problems that everyone who expects to pass the course should be able to do. The second s
Applied Combinatorics
Preliminary Edition
January 5, 2014
Mitchel T. Keller
Washington & Lee University
William T. Trotter
Georgia Institute of Technology
Copyright c 2014 Mitchel T. Keller and William T. Trotter.
This text is licensed under the CreativeC
Math 3012 Midterm 1, Fall 2016
September 30, 2016
1. How many strings of length 10 are there meeting the following restrictions?: the first 5 symbols are chosen from the 26 letters a through
z (lowercase), and the letters come in alphabetical order (e.g.
MATH 3012 C Quizzes
February 11, 2017
01/11 What portion of the final grade is each quiz worth assuming there are 40
quizzes in total? 0.25%. (Note: There will be around 35 quizzes in total,
so each quiz is actually worth 0.285% ).
01/13 How many binary s
Strong Mathematical induction
Principle
Suppose S is a statement about natural numbers. If
I
S is true for 1 and
I
whenever S is true for 1, 2, 3, . . . , k, then S is true for k + 1;
then S is true for every natural number.
1/6
Recall (Fibonacci sequence
Combinatorial proofs revisited
1/8
Combinatorial proofs
Exercise
Let n N. Explain why 1 + 3 + 5 + 7 + + 2n 1 = n2 .
2/8
Combinatorial proofs
Exercise
Let n N. Explain why
n
0
+
n
1
+
n
2
+ +
n
n
= 2n .
3/8
Combinatorial proofs
Exercise
Let n N. Explain wh
1/7
Counting solutions
Exercise
Determine the number of solutions to
x1 + x2 + x3 + x4 = 15,
subject to x1 , x2 , x3 , x4 Z, x1 , x2 , x3 0 and 0 x4 9.
2/7
Theorem (Binomial Theorem)
Let x, y R with x + y 6= 0. Then for each n N we have
n
X
n nk k
(x + y
Applied Combinatorics
Preliminary Edition
February 15, 2015
Mitchel T. Keller
Washington & Lee University
William T. Trotter
Georgia Institute of Technology
c 20082015 Mitchel T. Keller and William T. Trotter.
Copyright
This text is licensed under the Cr
Math 3012 Lecture Notes Sets
August 27, 2013
Definition. A set is an unordered collection of objects (notation: A = cfw_ a, b, c, d ). The
objects in A are called its elements or members. We write a A to denote that a is an
element of A.
Definition. The c
Math 3012 Lecture Notes Matchings
November 13, 2013
Recall that we used the Max Flow-Min Cut theorem to show
Theorem. The maximum matching in a bipartite graph G = (V1 V2 , E) has size
|V1 | max(|S| |N (S)| ,
SV
where N (S) = cfw_ v V (G) : v u for some u
Math 3012-N2 Practice Final
The final will be administered in Weber SST Lecture Hall 2, 8am-10:50am Monday December
9 2013. You will be allowed to bring to the final a single (double-sided) 8.5 11 inch cheat sheet
of reference notes. You must create this
Math 3012-N2 Homework II
due Friday, September 13 2013 at 9:05 am
1. (K-T 2.11) A donut shop sells 12 types of donuts. A manager wants to buy six donuts, one each for
himself and his five employees.
(a) Suppose he does this by selecting a specific type of
Euclidean Algorithm
Theorem
Let m, n be positive integers and let q and r be the unique integers
for which
m = q n + r and 0 r < n.
If r > 0, then gcd(m, n) = gcd(n, r ).
1/9
Recursive gcd
Algorithm
1: def rgcd(m,n):
2:
r = m0
3:
if r=0:
4:
return n
5:
el
Combinatorial problems solved recursively
Exercise (ad-free words)
Consider alphabet A = cfw_a, b, c, d, e. Find a recursive formula for
the number of words over A that do not contain ad as a substring
and have length n.
1/6
Well-ordering principle
Princi
Midterm 1, Math 3012, Fall 2014
September 19, 2014
Instructions: Show All your work if you want full credit. Do not use
programmable calculators.
1. Determine the number of lattice walks from (2, 3) (10, 11), given
that at each step in the walk you are on
Midterm 1, Math 3012, Fall 2009
September 24, 2009
1. Define the following terms.
a. Bayess Rule (give precise formula).
b. De Morgans laws for sets.
c. Rule of the sum.
d. Universal quantifier.
e. Symmetric difference.
2. Consider the string AAAABBBCC. C
Midterm 1, Math 3225, Fall 2010
October 13, 2010
1. Define the following terms.
a. The three Kolmogorov axioms of probability (list them).
b. sigma algebra.
c. Binomial distribution (write out the pdf, given n and p).
d. Define what it means for the event
Math 3012 - K
Spring 2017
Homework 2
2/25/2017
Solution
Name:
This solution contains 3 pages (including this cover page) and 3 questions.
Total of points is 30.
Three problems (30 points) will be graded on accuracy and remaining problems will be
graded on
Midterm 1, Math 3012, Fall 2014
September 23, 2014
Instructions: Show All your work if you want full credit. Do not use
programmable calculators.
1. Determine the number of strings of length 10 consisting of 0s, 1s and
2s, such that the number of 1s excee
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Applied Combinatorics
Applied Combinatorics
Mitchel T. Keller
Washington and Lee University
Lexington, Virginia
William T. Trotter
Georgia Institute of Technology
Math 3012C
Date: February 17, 2017
Midterm 1
Time: 10:05 to 10:55
Spring 2017
Duration of exam: 50 min
First Name (Print):
Last Name (Print):
This exam contains 1 page (including this cover page) and 7 questions. Total of points is 100.
Instructions
1. Pl
Exercise
Suppose that the elements of [36] are painted on a roulette in a
random order. Show that there exist three consecutive numbers on
the roulette whose sum is at least 55.
28 15 36 5
2 3
6
29
34
14
35
10
25
24
12
30
11
17
13 1
7 4 33
22
19
27
20
9
8
Distributing balls
Exercise
You are assigned the task of distributing n (indistinguishable) balls
into k (different) containers. In how many ways can this be
accomplished?
Hint
Try encoding the way of distributing the balls using a binary string.
What is
Multinomial coefficients
Numbers of the form
n
k1 , k2 , k3 , . . . , kr
=
n!
k1 !k2 ! . . . kr !
are called multinomial coefficients.
Example
We are given n balls labeled 1, 2, . . . , n and we are asked to paint
k1 of them red, k2 blue and k3 = n (k1 +
Counting strings
Exercise (Canadian Postal Codes)
Canadian postal codes follow the format LDL DLD where L is a
letter and D is a digit. None of the letters D, F, I, O, Q or U are
allowed and also the postal code may not begin with W or Z.
Assuming 900,000
Exercises from textbook.
Chapter 2: 30.
Chapter 3: 4, 6, 10, 12.
1.
Prove that the sum of the interior angles of a convex nn-gon
is (n2)180(n2)180 for all n3n3.
30.
Determine the coefficient on x12y24 in (x3 + 2xy2 + y + 3)18. (Be careful, as x and y now
14. An ice cream shop has a special on banana splits, and Xing is taking advantage of it. Hes
astounded at all the options he has in constructing his banana split:
He must choose three different flavors of ice cream to place in the asymmetric bowl the
ba