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 _ _ _ _ _ = 105 +
1st 2 is 2nd digit: _ 2 _ _ _ _ = (9)(104)
1.1
14. a) r q
b) p q r
c) r p
d) p q r
e) (p q) r
f) r (q p)
24. a) If I am to remember to send you the address, then you will have to send me an email message.
(This has been slightly reworded so that the tenses make more sense.) b) If you were born in
1.4
2. a) This is true, since there is an a in orange.
b) This is false, since there is no a in lemon.
c) This is false, since there is no a in true.
d) This is true, since there is an a in false.
6. The answers given here are not unique, but care must be
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 desired.
6. Under the hypotheses, we have c = as and d = bt
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 (universe of discourse) is the set of
integers.
c) cfw_x
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 inductive step, we want to show for each
that P(n) implies P(n
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 for the rst initial, then 25 choices for the second, if n
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 that P0 = 1,
since there is just one permutation of a
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, using separate sets of points; an
alternate form
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 of which is the winning one, so the answer is 1/C(n,6).
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). The number
of people a person knows is the degree of the
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) This is a tree since it is connected and has no simple cir
2.5
2. a) This set is countably innite. The integers in the set are 11, 12, 13, 14, and so on. We can list these
numbers in that order, thereby establishing the desired correspondence. In other words, the
correspondence is given by 1 11, 2 12, 3 13, and s
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 obtain
a + b = 2s + 2t = 2(s + t). Since this represents
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 kind is allowed.
1. (15 points) In the chart below, place
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)(7)(4) = 168
3. How many three digit numbers with no r
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_ UNI_
Multiple Choice (2 points each)
1. If 

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 added the edges to the spanning tree.
B
2
6
A
5
C
7
6
A
10
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. (1 point) Write your full name and UNI: _
2. (4 points
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 kind is allowed.
1. (15 points) In the chart below, place
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 are rolled.
a. Find the probability that their sum is les
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 _ _ _ _ _ = 105 +
1st 2 is 2nd digit: _ 2 _ _ _ _ = (9)(104)
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.
Someone wants to be defeated.
At least one person want
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 numbers (with no leading zeroes) are the same when the o
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 creepy.
b. If you attend class, then you will learn and gr
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 you will learn and grow.
c. p q
2. (5 pts) Use a truth
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 to find the coefficient of x12 in the expansion of (1 +