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# DISCRETE MATHEMATICS MTH 355

OSU

GPA 3.79

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This 9 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 355 at Oregon State University taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/224443/mth-355-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15

Mth 355 Exam I Name ID Bent Petersen 355f2001examtex Dec 3 2001 Time 110 minutes This test is open book You may use any books or notes that you wish to use You may use a calculator Problem 1 20 points correct 0 points if wrong Let X be a set of cardinality 7 and let Y be a set of cardinality 5 The number of ways of partitioning X into 5 nonempty subsets is 140 Determine the number of epimorphisms surjections ontomaps sz A A 120 B 140 C 5040 16800 None of the foregoing HLetter corresponding to your answer to problem 1 Problem 2 20 points correct 0 points wrong Let A be the power set of the empty set Z and let B be the power set of A The cardinality of B is A 0 B 1 C 2 00 None of the foregoing HLetter corresponding to your answer to problem 2 Problem 3 20 points if correct 0 points if wrong In a class of 19 good students 6 students will be assigned a grade of A 11 students will be assigned a grade of B and the remainder will be assigned a grade of C How many ways are there to assign the grades A 6859 B 27132 C 2116296 D 1162261467 E None of the foregoing HLetter corresponding to your answer to problem 5 Problem 4 20 points if correct 0 points if wrong An urn contains a large number of red white and blue marbles How many ways are there of selecting 9 marbles from the urn if the number of red marbles must be odd but not 3 and no more than 7 and if the number of white marbles must be even but not 4 and no more than 6 Hint Consider the generating function gz z I5 17 l 12 15l 12 I4 15 18 A 7 B 8 C 9 10 None of the foregoing HLetter corresponding to your answer to problem 4 Problem 5 20 points if correct 0 points if wrong The generating function for a certain sequence anngt0 is 9 I 17621714 Find a5 A 2 B 3 C 4 5 None of the foregoing HLetter corresponding to your answer to problem 5 Problem 6 20 points if correct 0 points if wrong Solve the recurrence relation an Zan1 2 a0 1 Then compute ago A 20 B 1048576 C 20971520 D 22020096 HLetter corresponding to your answer to problem 6 None of the foregoing Problem 7 20 points if correct 0 points if wrong Solve the recurrence relation an 4an2 2 a0 la1 0 Then compute am A 1025 B 1363 C 1365 D 1385 El HLetter corresponding to your answer to problem 7 None of the foregoing Problem 8 20 points if correct 0 points if wrong The adjacency matrix of a simple graph is always singular B invertible C symmetric HLetter corresponding to your answer to problem 8 nonsquare None of the foregoing Additional test policies for this class are provided on my web page httpwwwonid orst edu peterseb Use this space for scratch work Please do not Write in the boxes to the 1 Letter Grade 1 D This test only i D Cummulative right They are for your grades Do not be con cerned if there are 6 7 8 9 310 1 1 1 1 1 1 1 1 1 1 more boxes than i 3 i 3 i 1 problems Total Note There are 8 problems for a total of 160 points TakeHome Exam Due Thursday March 20 2003 330 PM or earlier Bent Petersen 355w2003examtex March 16 2003 You may use any books notes calculators computers whatnot Your solutions should be submitted by email with or without attachments as appropriate and should include at least a sketch of your work Part of the test is to gure out how to communicate your solutions You may nd a Maple Worksheet attachment convenient The last 2problems are Maple problems The solutions must be submitted on a Maple worksheet as an email attachment Your email must include the statement I certify that I have not received any help from anyone other than possibly from Bent Petersen for any of the work submitted for this exam Moreover I certify that I have not provided any help to anyone else nor engaged in any discussion of the content of the exam quot Problem 1 20 points Find the number of sequences j1j2j3j4 with 1Sj1 j2 ltJ393SJ394S128 Note the inequalities carefully Problem 2 20 points For a smooth realvalued function f of n real variables the theorem on the equality of mixed partial derivatives states that the partial derivatives do not depend on the order in which we differentiate but just on the number of times we differentiate relative to each variables There are no other restrictions on the values of the partial derivatives Given these facts how many distinct partial derivatives of order S m are generally possible for a smooth function of n real variables Problem 3 20 points How many integers between 37 and 20349 are divisible by 4 and by 6 but not by 5 Problem 4 20 points Let d7 be the number of derangements of n distinct objectsl Find the generating function for the sequence dn Hi Problem 5 20 points An urn contains 6 red marbles 4 green marbles 7 white marbles and 5 blue marbles ln how many ways can we select a set 12 marbles containing an odd number of white marbles Problem 6 20 points Solve the recurrence relation an 2an 71 an 7 2 7 2an 7 3 a01al 3a2 71 Problem 7 20 points Solve the recurrence relation an 2an 7 l 7 4an 7 2 8an 7 3 a0 4al 8a2 0 For what values of n is an 0 Problem 8 20 points We may think of the n cube as being constructed from two copies of the n7 lcube by joining the 2 1 vertices of the one copy with the corresponding vertices of the other copy by new edges Find a recurrence relation for the number an of edges of the ncube Noting al l solve for an Problem 9 20 points Maple Problem Write a Maple procedure funmf7A Which takes as argu ments a function f and a diagonizable square matrix A of any size and returns de ned as follows If SAS l D is diagonal7 say D diag17 7 An7 then diagf17 7 and is de ned to be S 1fDSr Use DjordanA775 from the linalg package to nd D and Sr Verify your work by comparing A3 and funmzigt 137A for some symmetric matrix A1 Then compute the absolute value of the matrix 2 l A l 1 72 l 7 that is7 compute funmabs7Ar Problem 10 20 points Maple Problem Write a Maple procedure pts Which given any number of lists L17 7 Ln of arbitrary lengths returns a list L such that the 16 entry of L is L1k7L2k7 7Lnk7 Where Is 1727 7N Where N is the minimum length of the Lkr Note pts172737475767778797 10711712 should return 17477710 275787117 37679712 and pts172737475767778710711712l should return 17477710 275787111 lf L17 7 Ln are lists of the same length What is the result of ptsopptsL17 7 Ln Mth 355 Midterm Name ID Bent Petersen 355w2003midtermtex Feb 26 2003 Time 110 minutes You may use one 85 X 11 inch sheet of notes You may use a calculator Problem 1 20 points Consider an m X n rectangular grid How many routes are there from the lower left corner to the upper right corner if we allow only travel to the right or upward Hint Each route may be associated with a string consisting of m Uls and n R s where R indicates a right step and U indicates an upward step Problem 2 20 points If we toss a fair coin 20 times we can record the outcome as a sequence of Hls for heads and T s for tails Clearly there are 220 possible strings What fraction of these strings contain exactly 10 Hls Problem 3 20 points Let X be a set of cardinality 12 and let Y be a set of cardinality 5 The number of partitions of X into 5 nonempty subsets is 51275 173797400 How many epimorphisms sz A Y are there Problem 4 20 points Find the number of solutions in nonnegative integers to the equation 11I213I442 Problem 5 20 points Find the number of solutions in nonnegative odd integers to the equation I1I213I442 Problem 6 20 points Find the number of solutions in positive integers to the inequality I1I213I4S36 Problem 7 20 points Find the number of solutions in nonnegative integers to the equation 11 12 13 10 if5 11 91 12 671322isodd Problem 8 20 points The generating function for a certain sequence a nzo is 1 W m Find us Problem 9 20 points Solve the recurrence relation an 6 1771 7 9 1772 With the initial conditions a0 2 a1 3 Problem 10 20 points Solve the recurrence relation an San1 4n3 With the initial condition a0 6 Additional test policies for this Class are provided on my web page httpwwwonid orst edu petersebi Please do not Write in the boxes to the right They are for your grades Do not be con cerned if there are than more boxes problems Use this space for scmtch work 1 Letter Grade 1 D This test only 1 D Cummulatz39ve 11 12 13 14 15 16 17 18 9 310 Total Note There are 10 problems for a total of 200 points Mth 355 Midterm I Name ID Bent Petersen 355w2003midtermsampletex Feb 26 2003 Time 110 minutes You may use one 85 X 11 inch sheet of notes You may use a calculator Problem 1 20 points How many strings are there of length 12 consisting of 3 A s7 5 B s and 4 cs Problem 2 20 points Let X be a set of cardinality 5 and let Y be a set of cardinality 12 How many monomorphisms sz A Y are there Problem 3 20 points Find the number of solutions in positive integers to the equation 11I213I442 Problem 4 20 points Find the number of solutions in nonnegative integers to the equation 11I213I4 42 if1122and1225 Problem 5 20 points Find the number of solutions in nonnegative even integers to the equation 11I213I442 Problem 6 20 points Find the number of solutions in nonnegative integers to the equation 11I213I4 42 if 11 and 12 are even and 13 and 14 are odd Problem 7 20 points Find the number of solutions in nonnegative integers to the inequality I1I213I4S36 Problem 8 20 points Find the number of solutions in nonnegative integers to the equation 11I213I412 if5 11 91 12 671322isprimeandz4 isodd Problem 9 20 points The generating function for a certain sequence a nzo is 1 ya W Find a3 Problem 10 20 points Find the characteristic polynomial and characteristic roots of the recurrence relation an 7an1 710an72 Problem 11 20 points Solve the recurrence relation an 2 anil 3 047172 With the initial conditions a0 87 a1 74 Problem 12 20 points Solve the recurrence relation an 4 1772 With the initial conditions a0 2 a1 8 Problem 13 20 points You deposit 50 per month into an initially empty bank account Suppose you earn percent on the account at the end of each month Find a recurrence equation for the account balance In at the end of the nth month What is your balance at the end of 24 months Problem 14 20 points Solve the recurrence relation an 2047171 372 n2 3 With the initial conditions a0 2 a1 3 Problem 15 20 points An urn contains a large number of red7 White and blue marbles How many ways are there of selecting 10 marbles from the urn if the number of red marbles must be odd and at least 37 if the number of White marbles must be even7 but no more than 4 and the number of blue marbles must be prime

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