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Mathematics
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Nov 24, 2024
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MODULE THREE PROBLEM SET
This document is proprietary to Southern New Hampshire University. It and the problems within may not be posted on any non-SNHU website.
Your Name Here
1
Directions: Type your solutions into this document and be sure to show all steps
for arriving at your solution. Just giving a nal number may not receive full credit.
Problem 1
A 125-page document is being printed by ve printers. Each page will be printed exactly once.
(a)
Suppose that there are no restrictions on how many pages a printer can
print. How many ways are there for the 125 pages to be assigned to the
ve printers?
One possible combination: printer A prints out pages 2-50, printer B prints
out pages 1 and 51-60, printer C prints out 61-80 and 86-90, printer D prints
out pages 81-85 and 91-100, and printer E prints out pages 101-125.
(b)
Suppose the rst and the last page of the document must be printed in
color, and only two printers are able to print in color. The two color print-
ers can also print black and white. How many ways are there for the 125
pages to be assigned to the ve printers?
(c)
Suppose that all the pages are black and white, but each group of 25
con-secutive pages (1-25, 26-50, 51-75, 76-100, 101-125) must be
assigned to the same printer. Each printer can be assigned 0, 25, 50, 75,
100, or 125 pages to print.
How many ways are there for the 125 pages to be assigned to the ve printers?
Problem 2
Ten kids line up for recess. The names of the kids are:
fAlex, Bobby, Cathy, Dave, Emy, Frank, George, Homa, Ian, Jimg.
Let S be the set of all possible ways to line up the kids. For example, one order
might be:
(Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby)
The names are listed in order from left to right, so Frank is at the front of the
line and Bobby is at the end of the line.
Let T be the set of all possible ways to line up the kids in which George is
ahead of Dave in the line. Note that George does not have to be immediately
ahead of Dave. For example, the ordering shown above is an element in T .
Now de ne a function f whose domain is S and whose target is T . Let x be an
element of S, so x is one possible way to order the kids. If George is ahead of Dave
in the ordering x, then f(x) = x. If Dave is ahead of George in x, then f(x) is the
ordering that is the same as x, except that Dave and George have swapped places.
(a)
What is the output of f on the following input?
(Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby)
(b)
What is the output of f on the following input?
(Emy, Ian, Dave, Homa, Jim, Alex, Bobby, Frank, George, Cathy)
(c)
Is the function f a k-to-1 correspondence for some positive integer k? If so, for what value of k? Justify your answer.
(d)
There are 3628800 ways to line up the 10 kids with no restrictions on
who comes before whom. That is, jSj = 3628800. Use this fact and the
answer to the previous question to determine jT j.
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Problem 3
Consider the following de nitions for sets of characters:
Digits = f0; 1; 2; 3; 4; 5; 6; 7; 8; 9g
Letters = fa; b; c; d; e; f; g; h; i; j; k; l; m; n; o; p; q; r; s; t; u; v; w; x; y; zg Special characters = f ; &; $; #g
Compute the number of passwords that satisfy the given constraints.
(i)
Strings of length 7. Characters can be special characters, digits, or letters, with no repeated characters.
(ii)
Strings of length 6. Characters can be special characters, digits, or
letters, with no repeated characters. The rst character can not be a
special char-acter.
Problem 4
A group of four friends goes to a restaurant for dinner. The restaurant o ers 12
di erent main dishes.
(i)
Suppose that the group collectively orders four di erent dishes to share.
The waiter just needs to place all four dishes in the center of the table.
How many di erent possible orders are there for the group?
(ii)
Suppose that each individual orders a main course. The waiter must re-
member who ordered which dish as part of the order. It’s possible for
more than one person to order the same dish. How many di erent
possible orders are there for the group?
How many di erent passwords are there that contain only digits and lower-
case letters and satisfy the given restrictions?
(iii)
Length is 7 and the password must contain at least one digit.
(iv)
Length is 7 and the password must contain at least one digit and at least one letter.
Problem 5
A university o ers a Calculus class, a Sociology class, and a Spanish class. You are given data below about two groups of students.
(i)
Group 1 contains 170 students, all of whom have taken at least one of
the three courses listed above. Of these, 61 students have taken
Calculus, 78 have taken Sociology, and 72 have taken Spanish. 15 have
taken both Cal-culus and Sociology, 20 have taken both Calculus and
Spanish, and 13 have taken both Sociology and Spanish. How many
students have taken all three classes?
(ii)
You are given the following data about Group 2. 32 students have taken
Calculus, 22 have taken Sociology, and 16 have taken Spanish. 10 have
taken both Calculus and Sociology, 8 have taken both Calculus and
Span-ish, and 11 have taken both Sociology and Spanish. 5 students
have taken all three courses while 15 students have taken none of the
courses. How many students are in Group 2?
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Problem 6
A coin is ipped ve times. For each of the events described below, express the
event as a set in roster notation. Each outcome is written as a string of length 5
from fH; T g, such as HHHT H. Assuming the coin is a fair coin, give the proba-
bility of each event.
(a) The rst and last ips come up heads.
(b)
There are at least two consecutive ips that come up heads.
(c)
The rst ip comes up tails and there are at least two consecutive ips that come up heads.
Problem 7
An editor has a stack of k documents to review. The order in which the doc-
uments are reviewed is random with each ordering being equally likely. Of the k
documents to review, two are named \Relaxation Through Mathematics" and
\The Joy of Calculus." Give an expression for each of the probabilities below as
a function of k. Simplify your nal expression as much as possible so that your
answer does not include any expressions in the form
a b
.
(a)
What is the probability that \Relaxation Through Mathematics" is rst to review?
(b)
What is the probability that \Relaxation Through Mathematics" and \The Joy of Calculus" are next to each other in the stack?