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这是加州理工学院组合学课程的代写成功案例。
Math121a课程简介
Math 121 is our standard three-term course on combinatorics though we are only able to offer two terms this year. The first term covers “structural” combinatorics, the study of interesting discrete structures, both in terms of general properties as well as in terms of the existence or classification of particularly nice or extreme instances. We will spend much of the term on graphs and related topics such as Ramsey theory, with the remainder of the term spent covering other important structures Latin squares, codes, designs, finite geometries, etc.. The second term will cover more quantititative aspects of combinatorics, as well as applications to and from algebra.
Prerequisites
Grades: Grades will be assessed on the basis of homeworks, which will be given roughly biweekly. There will be no exams (or midterms). Students may discuss problems with each other but have to write down the solutions independently.
The following restrictions hold for all problems of the set: Students may ask questions to the TA (if one exists) and the professor. Resources which you may use while working on the homework include any books and non-interactive websites (i.e. no posting of the questions on internet fora). For calculations which you can do by hand, you may use a computer algebra program Mathematica, Maple, etc.; you may also fell free to use a computer to gain intuition say by solving small instances of the problems, so long as your eventual proof does not depend on such a computation. If you use significant ideas from any source, other than the books mentioned on this website, you should mention where you got them from.
Math121a Combinatorics HELP（EXAM HELP， ONLINE TUTOR）
How many different three digit positive integers are there? (No leading zeroes are allowed.) How many positive integers with at most three digits? What are the answers when “three” is replaced by ” $n ? “$
1.1.2. A small neighboring country of the one we revisited in Example 1.3 has the same alphabet and the same rules of formation, but names are only five letters long. How many names are possible?
1.1.3. Prove that the number of subsets of a set $S$, including the empty set and $S$ itself, is $2^{|S|}$. Hint. For each element of $S$ you must make one of two choices: ” $x$ is/isn’t in the subset.”
A composition of a positive integer $n$ is an ordered list of positive integers (called parts) that sum to $n$. The four compositions of 3 are $3 ; 2,1 ; 1,2$ and $1,1,1$.
(a) By considering ways to insert plus signs and commas in a list of $n$ ones, obtain a formula for the number of compositions of $n$.
Hint. The four compositions above correspond to $1+1+1 ; 1+1,1 ; 1,1+1$ and $1,1,1$, respectively.
(b) Prove that the average number of parts in a composition of $n$ is $(n+1) / 2$.
Hint. Reverse the roles of “+” and “,” and then look at the number of parts in the original and role-reversed compositions.
In Example 1.3 we found that there were 648 possible names. Suppose that these are listed in the usual dictionary order. What is the last word in the first half of the dictionary (the 324th word)? the first word in the second half?
1.2.1. Find to two decimal places the answer to the birthday question asked in Example 1 (p. 1). Hint. Assigning birthdays to 30 people is the same as forming an ordered list of 30 dates.
1.2.2. Use (1.2) to estimate the solution to the birthday problem in Example 1 (p. 1).
1.2.3. How many ways are there to form an ordered list of two distinct letters from the set of letters in the word COMBINATORICS? three distinct letters? four distinct letters?
1.2.4. Repeat the previous problem when the letters need not be distinct but cannot be used more often than they appear in COMBINATORICS.
We are interested in forming 3 letter words (“3-words”) using the letters in LITTLEST. For the purposes of the problem, a “word” is any ordered list of letters.
(a) How many words can be made with no repeated letters?
(b) How many words can be made with unlimited repetition allowed?
(c) How many words can be made if repeats are allowed but no letter can be used more often than it appears in LITTLEST?
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