Name_
COMS 3203
Exam 3
Spring 2013
No calculators, notes, textbooks. Seriously, dont cheat.
Write your name on every page or that page will not be graded.
1. (4 points each) Two fair sixsided dice ar
COMS 3203
Exam 2
March 28, 2012
No notes, books, calculators, or cheating allowed.
1. (6 points each)
a. How many positive integers less than 1 million contain the digit 2?
1st 2 is 1st digit: 2 _ _ _
5.1
3
2
4. a) Plugging in n = 1 we have that P(1) is the statement 1 = [1 (1 + 1)/2] .
b) Both sides of P(1) shown in part (a) equal 1.
c) The inductive hypothesis is the statement
d) For the inductiv
6.1
2. By the product rule there are 27 37 = 999 oces.
4. By the product rule there are 12 2 3 = 72 dierent types of shirt.
6. By the product rule there are 4 6 = 24 routes.
8. There are 26 choices fo
8.1
2. a) A permutation of a set with n elements consists of a choice of a first element (which can be done in
n ways), followed by a permutation of a set with n 1 elements. Therefore Pn = nPn1 . Note
9.1
2. a) (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (5,5),
(6,6)
b) We draw a line from a to b whenever a divides b, us
7.1
2. The probability is 1/6, since there are six equally likely outcomes.
24. In each case, if the numbers are chosen from the integers from 1 to n, then there are C(n,6) possible
entries, only one
10.2
18. This is essentially the same as Exercise 40 in Section 6.2, where the graph models the know each
other relation on the people at the party. See the solution given for that exercise (below). T
11.1
2. a) This is a tree since it is connected and has no simple circuits.
b) This is a tree since it is connected and has no simple circuits.
c) This is not a tree, since it is not connected.
d) Thi
8.4
14. Each child will correspond to a factor in our generating function. We can give 0, 1, 2,
or 3 figures to the child; therefore the generating function for each child is 1 + x + x2 + x3.
We want
COMS W3261
Computer Science Theory Section 002
Final Review
The CS Theory Final
The final will be held on Tuesday, December 20, 2016, 4:107:00 pm.
The sample problems below are intended to help you
Please read this assignment carefully and follow the instructions
EXACTLY.
Submission

Please refer to the lab retrieval and submission instruction, which outlines
the only way to submit your lab as
Practice Midterm Exam
1. Consider the differential equation
t2 y 00 + 2ty 0 2y = 0,
t > 0.
Using that y1 (t) = t is a solution of this equation, use the method of reduction of order to
find a second l
Practice Midterm Exam
1. Consider the differential equation
t2 y 00 + 2ty 0 2y = 0,
t > 0.
Using that y1 (t) = t is a solution of this equation, use the method of reduction of order to
find a second l
CS5371
Theory of Computation
Lecture 15: Computability VI
(Post
s Problem, Reducibility)
Objectives
In this lecture, we introduce Post
s
correspondence problem (playing with a
special type of domino)
2.1
2. There are of course an infinite number of correct answers.
a) cfw_ 3n  n = 0, 1, 2, 3, 4 or cfw_ x  x is a multiple of 3 0 x 12 .
b) cfw_ x  3 x
3 , where we are assuming that the domain (u
4.1
4. Suppose a  b, so that b = at for some t, and b  c, so that c = bs for some s. Then substituting the first
equation into the second, we obtain c = (at)s = a(ts). This means that a  c, as desi
Name_
Answer questions 13 using only digits from the set cfw_0, 1, 2, 3, 4, 5, 6.
1. How many three digit numbers can be formed?
(6)(7)(7) = 294
2. How many even three digit numbers can be formed?
(6
COMS 3203
Final Exam
Spring 2014
No cheating allowed.
Assume all graphs are simple unless otherwise noted.
Write your answers on the FRONT pages only. Work done on the reverse pages will NOT be graded
Name_
COMS 3203 Exam 2
Spring 2013
SOLUTION
1. (5 points) Find the minimum spanning tree for the following graph using either Prims or Kruskals algorithm. You
must indicate the order in which you adde
COMS 3203
Exam 2
Spring 2014
No cheating allowed.
Assume all graphs are simple unless otherwise noted.
Write your answers on the FRONT pages only. Work done on the reverse pages will NOT be graded.
1.
Name_
COMS 3203
Final Exam
Spring 2012
Write your name on every page. If you do not do this, I will deduct 5 points! This exam is
closed book. No notes, calculators, cell phones, or cheating of any ki
COMS 3203
Exam 2
March 28, 2012
No notes, books, calculators, or cheating allowed.
1. (6 points each)
a. How many positive integers less than 1 million contain the digit 2?
1st 2 is 1st digit: 2 _ _ _
Name_
COMS 3203
Exam 1
Spring 2013
No calculators, notes, textbooks, or cheating of any kind.
1. (5 points each) Write the negation of each of the following statements.
a. No one wants to be defeated.
NAME_
COMS 3203
Exam 2
March 28, 2012
No notes, books, calculators, or cheating allowed.
1. (6 points each)
a. How many positive integers less than 1 million contain the digit 2?
b. How many 5digit n
Name_
COMS 3203
Exam 1 Solutions
February 15, 2012
No notes, books, calculators, or cheating allowed.
1. (3 pts each) Negate the following statements.
a. Some clowns are not creepy.
All clowns are cre
Name_
COMS 3203
Exam 1
February 15, 2012
No notes, books, calculators, or cheating allowed.
1. (3 pts each) Negate the following statements.
a. Some clowns are not creepy.
b. If you attend class, then
Name_
COMS 3203
Final Exam
Spring 2012
Write your name on every page. If you do not do this, I will deduct 5 points! This exam is
closed book. No notes, calculators, cell phones, or cheating of any ki
1.7
2. We must show that whenever we have two even integers, their sum is even. Suppose that a and b are
two even integers. Then there exist integers s and t such that a = 2s and b = 2t. Adding, we ob