1 The Foundations: Logic And Proofs 2 Basic Structures: Sets, Functions, Sequences, Sums, And Matrices 3 Algorithms 4 Number Theory And Cryptography 5 Induction And Recursion 6 Counting 7 Discrete Probability 8 Advanced Counting Techniques 9 Relations 10 Graphs 11 Trees 12 Boolean Algebra 13 Modeling Computation A Appendices expand_more
6.1 The Basics Of Counting 6.2 The Pigeonhole Principle 6.3 Permutations And Combinations 6.4 Binomial Coefficients And Identities 6.5 Generalized Permutations And Combinations 6.6 Generating Permutations And Combinations Chapter Questions expand_more
Problem 1RQ: Explain how the sum and product rules can be used to find the number of bit strings with a length... Problem 2RQ: Explain how to find the number of bit strings of length not exceeding 10 that have at least one o... Problem 3RQ Problem 4RQ: How can yon find the number of possible outcomes of a playoff between hvo teams where the first team... Problem 5RQ: How can you find the number of bit strings oflength ten that either begin with 101 or end with 010? Problem 6RQ: State the pigeonhole principle, Explain how the pigeonhole principle can be used to show that among... Problem 7RQ: State the generalized pigeonhole principle. Explain how the generalized pigeonhole principle can be... Problem 8RQ: ft What is the difference between an r-combination and an r-permutation of a set withnelements?... Problem 9RQ: What i s Pas cal's tri angle? How can arow of Pascal's triangle be produced from the one above it? Problem 10RQ: What is meant by a combinatorial proof of an identity? How is such a proof different from an... Problem 11RQ: ii. Explain how to prove Pascal's identity using a combinatorial argument. Problem 12RQ: Stateth e bin omial th eor em. Explain how to pr o ve th e bin omial th eor em using a c... Problem 13RQ: Explain how to find a formula for the number of ways to select r objects from n objects when... Problem 14RQ: Letnand r be positive integers. Explain why the number of solutions of the equation xt+x2 + ¦¦• +... Problem 15RQ Problem 16RQ Problem 17RQ: a) How many ways are there to deal hands of five cards to six players from a standard 52-card... Problem 18RQ: Describe an algorithm for generating all the combinations of the set of thensmallest positive... Problem 1SE: i. How many ways are there to choose 6 items from 10 distinct items when the items in the choices... Problem 2SE: a.H 01 v many ways ar e ther e to ch o o se1o items fr om 6distinct items when the items in the... Problem 3SE Problem 4SE: How many strings of length10either start with ooo or end with 1111? Problem 5SE Problem 6SE Problem 7SE Problem 8SE: Hoi v many positive integers less than iqoo have exactly three decimal digits? have an odd number of... Problem 9SE Problem 10SE Problem 11SE Problem 12SE: How many people are needed to guarantee that at least two were born on the same dav of the week and... Problem 13SE: Show that given anv set of 10 positive integers not exceeding 50 there exist at least two different... Problem 14SE Problem 15SE Problem 16SE Problem 17SE: Show that in a sequence ofmintegers there exists one or more consecutive terms with a sum divisible... Problem 18SE Problem 19SE: Show that the decimal expansion of a rational number must repeat itself from some point onward Problem 20SE: Once a computer worm infects a personal computer via an infected e-mail message, it sends a copy of... Problem 21SE: si.How many ways are there to choose a dozen donuts from 20 varieties if there are no two donuts of... Problem 22SE: ss.Findn if P(n,2] = 110. J\?i, n] = 5040. e)P(n,4] = liPtTi, 2) Problem 23SE Problem 24SE: Show that ifnandrare nonnegative integers and n > r, then /*«« + l.r) =Pin,r>l« + I>/(« + I - r(. Problem 25SE Problem 26SE: Give a combinatorial proof ofCorollary 2ofSection 6,4by setting up a correspondence between the... Problem 27SE Problem 28SE: a8. Prove using mathematical induction that O>• 2) = c Problem 29SE Problem 30SE: Show that V7' XIt. I = (’) if nis an integer with Problem 31SE Problem 32SE Problem 33SE: How many bit strings of length n, where n > 4, contain exactly two occurrences Problem 34SE Problem 35SE Problem 36SE Problem 37SE: How many ways are there to assign 24 students to five faculty advisors? Problem 38SE Problem 39SE: - How many solutions are there to the equation xt + x2+x3 = 17, where x^x^ and x3 are nonnegative... Problem 40SE: How many different strings can be made from the wordPEPPERCORNwhen all the letters are used? How... Problem 41SE: How many subsets of a set with ten el e m ents have fewer than five elements? have more than seven... Problem 42SE Problem 43SE Problem 44SE: How many ways are. there to seat six boys and eight girls in a row of chairs so that no two boys are... Problem 45SE: How many ways are there to distribute six objects to five boxes if both the objects and boxes are... Problem 46SE: How many ways are there to distribute five obj ects into six boxes if both the obj ects and boxes... Problem 47SE: Find these signless Stirling numb er s of the first kind a) c(4, 2] e) e(4,3)d)0(5,4] Problem 48SE: Show that ifnis a positive integer, then , Problem 49SE Problem 50SE Problem 51SE Problem 52SE: j2, How many n-element RXA sequences consist of 4 As, 3 Cs, 2 Us, and 2 Gs, and end with CAA?... Problem 53SE: *53. Suppose that when an enzyme that breaks RXA chains after each G linkis applied to a 12-link... Problem 54SE: Suppose that when an enzyme that breaks RXA chains after each G linkis applied to a 12-link chain,... Problem 55SE: Devise an algorithm for generating all the r-permutations of a finite set when repetition is... Problem 56SE: Devise an algorithm for generating all the r-combinations of a finite set when repetition is... Problem 57SE Problem 58SE Problem 1CP Problem 2CP Problem 3CP Problem 4CP Problem 5CP: Given a positive integern,listallthe permutations of the set{1, 2,3,n}in lexicographic order Problem 6CP Problem 7CP Problem 8CP Problem 9CP Problem 10CP Problem 1CAE Problem 2CAE Problem 3CAE Problem 4CAE Problem 5CAE Problem 6CAE Problem 7CAE Problem 8CAE Problem 9CAE Problem 1WP: Describe some of the earliest uses of the pigeonhole principle by Dirichlet and other... Problem 2WP Problem 3WP: Discuss the importance of combinatorial reasoning in gene sequencing and related problems involving... Problem 4WP: Jlanv combinatonal identities are described in this book, Find some sources of such identities and... Problem 5WP Problem 6WP Problem 7WP Problem 8WP: Describe the latest discoveries of values and bounds for Ramsev numbers. Problem 9WP Problem 10WP format_list_bulleted