Math 471
Group Work #8
Fall 2011
Problem 1.
(a): Find a recurrence relation for the number tn of bit strings of length n that contain
three consecutive zeros.
SOLUTION: We break up all of the bit strings of length n according to the following nonoverlappi
Math 471
Midterm I Information
SOLUTIONS TO SAMPLE PROBLEMS
Fall 2011
Problem 1:
(a): How many bit strings (strings of zeros and ones) of length 10 begin with 3 zeros or
end with two ones? How many bit strings of length 10 contain 8 or more consecutive
ze
Math 471
Midterm I Review Session Problems
SOLUTIONS
Fall 2011
Problem 1. A book publisher has 3000 textbooks to store in three warehouses. How
many ways are there to do this if
(a): the books are identical?
SOLUTION: Use stars and bars with 3000 stars an
Math 471
Midterm I Review Session Problems
SOLUTIONS
Fall 2011
Problem 1. A book publisher has 3000 textbooks to store in three warehouses. How
many ways are there to do this if
(a): the books are identical?
SOLUTION: Use stars and bars with 3000 stars an
October 4, 2004
Midterm I
Math 471
Name:
Instructions:
1. This test has 5 problems, totalling 100 points. Some problems have several parts.
2. No notes, books, or calculators.
3. You should not simplify numerical answers. You can leave answers in terms of
October 4, 2004
Midterm I
Math 471
SOLUTIONS
Name:
Problem 1. From a group of 7 men and 11 women, a committee of ocers is to be
formed.
(a): (5 points) How many committees can be formed with 10 ocers if there are no
restrictions?
SOLUTIONS: From the 18 to
Math 471
Group Work # 10
SOLUTIONS
Fall 2011
Problem 1. The graphs are not isomorphic. For instance, in the graph on the left, every vertex of
degree 3 (namely: a,d,e,h) is adjacent to another vertex of degree 3, but in the graph on the right,
vertex 7 ha
Math 471
Group Work # 9
SOLUTIONS
Fall 2011
Problem 1. For each pair of graphs below, decide whether or not they are isomorphic.
For those that are, give an explicit isomorphism between them. For those that are not,
nd a structural property that is not co
Math 471 Homework Solutions: Assignment # q
Chapter 7.
Problem- We guess a solution of the form h7, = q". Substituting this guess into the recurrence
relation yields q" = 4q"'2. Dividing through by qnz, we obtain q2 = 4. Hence, q =: 12. So the
general sol
Math 471
Group Work # 11
SOLUTIONS
Fall 2011
Problem 1.
(a): Prove that if G is a connected, planar graph with e > 1 and containing no 3-cycles,
then e 2n 4.
SOLUTION: If G is planar and contains no 3-cycles, then the degree of each region R in the planar
October 3, 2011
Midterm I
Math 471
Name:
Problem 1. How many strings of eight (not necessarily distinct) capital letters
(a): contain at least six Ts?
SOLUTION: We compute the number of strings with exactly six Ts, exactly seven Ts, and exactly
eight Ts,
Math 471
Midterm I Review Session Problems
Fall 2011
Problem 1. A book publisher has 3000 textbooks to store in three warehouses. How many ways
are there to do this if
(a): the books are identical?
(b): the books are distinguishable?
(c): the books are di
Math 471
Midterm I Information
Fall 2011
WHEN: Monday, October 3, in class (no notes, books, calculators)
COVERAGE: The midterm will cover the material discussed in lecture from Sections 2.12.5,
3.13.2, 5.15.2, 5.4, and 6.16.4.
EXTRA OFFICE HOURS:
Thursda
Math 471 Homework Solutions: Assignment # 2
Chapter 2.
Problem 2. We break down the choices according to the tasks necessary to order 52 cards with
all of the same suits together. There are 4! ways to decide what order to display the various suits
(since
Math 471
Group Work #1
SOLUTIONS
Fall 2011
Problem 1. Prove that at a party of n > 1 people, there are two people who have the
same number of acquaintances. [It is assumed that no one is acquainted with him or
herself.]
SOLUTION: The possible numbers of p
Math 471
Group Work #2
SOLUTIONS
Fall 2011
Problem 1. How many strings of seven distinct letters of the alphabet contain
(a): at least one vowel? [Hint: How many contain no vowels?]
SOLUTION: There are 21 non-vowels in the alphabet, so we have P (21, 7) =
Math 471 Homework Solutions: Assignment # 5
Chapter 6.
Problem 3. ANSWER: 9883 From the total of 10,000, we subtract those numbers that are either
a perfect square or a perfect cube.
Let A := cfw_ numbers in the range which are perfect squares
Let B := cf
Math 471 Homework Solutions: Assignment # 4
Chapter 2.
Problem 48.
SOLUTION #1: We prove this by induction on n. If n = 0, then we have m As to arrange, and
there is only 1 way to do this. But with n = 0, the given formula is C(m + 1, m + 1) = 1, which
co
Math 471
Group Work #3
Fall 2011
Problem 1.
(a): How many dierent strings can be made using all 10 letters of the word GOOGOLPLEX?
SOLUTION: Given 10 slots, we choose 2 slots in which to put the Gs: C(10, 2). Now there are 8
slots left, and we choose 3 sl
Math 471
Group Work #6
SOLUTIONS
Fall 2011
Problem 1. How many ways can you place six non-attacking rooks on the 6 6 chessboard below with forbidden positions as shown?
SOLUTION: In this problem, we have
r1 = 8,
r2 = 21,
r3 = 22,
r4 = 9,
r5 = 1,
r6 = 0.
T
Math 471
Group Work #4
SOLUTIONS
Fall 2011
Problem 1. Prove that for positive integers r m k, we have
r
m
m
k
=
rk
mk
r
k
,
by giving
(a): an algebraic proof.
SOLUTION: We have
r
m
m
k
=
=
m!
r!
r!
=
=
m!(r m)! k!(m k)!
(r m)!k!(m k)!
(r k)!
r!
=
k!(r k)!
Math 471
Group Work #7
SOLUTIONS
Fall 2011
RANDOM REVIEW PROBLEMS FOR MIDTERM #1
Problem 1. A student has 3 mangos, 2 papayas, and 2 kiwi fruits. If the student eats
one piece of fruit each day, and only the type of fruit matters, in how many dierent
ways
Math 471
Group Work #5
SOLUTIONS
Fall 2011
Problem 1. How many 5-card poker hands from a standard deck contain exactly 2
spades, at most 1 heart, or a ush1 ?
SOLUTION: Let
A = cfw_poker hands with exactly 2 spades,
B = cfw_poker hands with at most 1 heart
Math 471 Homework Solutions: Assignment # 1
Chapter 2.
Problem 4. Label n boxes as follows: cfw_1, 2, cfw_3, 4, cfw_5, 6, . . ., cfw_2n 1, 2n. We place each of the
n + 1 integers into which box has its label. Since there are n boxes and n + 1 integers, th