数学代写|MATH579 Combinatorics

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MATH579课程简介

Overview:
Combinatorics is concerned with the sizes of finite sets. For example, consider the set of all possible different necklaces made with $m$ beads, chosen from a tub of $n$ different beads. A variety of tools have been developed to find the size of such a set exactly, or perhaps to find an estimate or bound.
Textbook:
A Walk Through Combinatorics, by Miklós Bóna, 3rd edition (older editions are permissible, but contain fewer exercises and more errors). This course will cover chapters 1-8.2, omitting 6.2. There is also a supplement on recurrence relations, which is available on the instructor’s website. Students are expected to read the text; it is quite brief and easy to understand. It contains many exercises, some with brief solutions and some without.

Prerequisites 

Portfolio:
Students are expected to keep a portfolio in a three-ring binder or something similar, containing a detailed and complete solution to every exercise in the text (those marked + or ++ are optional). These portfolios will not be collected or checked, except upon a student’s request; however, they will be an invaluable resource in preparing for exams.

Students are NO’T’ required to personally solve every exercise appearing in their portfolios; they are strongly encouraged to collaborate with classmates. However, before accepting a classmate’s solution into their portfolio, students are expected to carefully check it for completeness and correctness.

Learning Objectives:
There are three distinct phases to solving a combinatorial problem. Generally, the first phase is the most difficult to learn, and the last phase is the easiest. Students will learn all three in this course. First, the problem must be categorized as to which combinatorial tool would be appropriate. Second, a model must be created that translates the abstract formulation of the problem into the symbols required for the combinatorial tools to work. ‘hird, the combinatorial tools must be applied to the symbols.

MATH579 Combinatorics HELP（EXAM HELP， ONLINE TUTOR）

For these next problems, a “word” is a string of letters, drawn from the 26 options ab… z. $n$ represents an arbitrary natural number; solve the problems for all $n$.

These are equivalent to lists of length $n$, drawn from [26]. Our answer is $26^n$.

If $n>26$, then the answer is 0 . Otherwise the answer is $26^{n}$. Note that our definition of falling powers does not define expressions like $26 \frac{30}{}$. This problem makes a good case that these should be defined to be zero.

If $n<26$ then the answer is 0 . Otherwise the answer is $26 ! S(n, 26)$.

The restriction reduces our 26 options to 21 ; our problem is equivalent to repeating problem 1 but drawing from $[21]$ instead of [26]. Our answer is $21^n$.

By the previous problem, $21^n$ words of length $n$ have no vowels. By a similar calculation, $5^n$ words of length $n$ have no consonants. All words of length $n$ either fall in one of the preceding, disjoint, categories, or else are of the type we want to count. Hence, $21^n+5^n+x=26^n$. We subtract, to find our desired $x=26^n-21^n-5^n$. This is valid for all $n \in \mathbb{N}$.

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