Posted on Categories:Combinatorics, 数学代写, 组合学

# 数学代写|组合学代写Combinatorics代考|CS-E4555 The sum and product rules

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|组合学代写Combinatorics代考|The sum and product rules

In this section we study two important results called the sum rule and the product rule, which demonstrate themselves in many combinatorial problems. We will see that they are nothing but the formulas for calculating the cardinalities of the union and the Cartesian product of finite sets. We introduce these rules by considering simple model problems.

Coffee-time browsing

• www.math.csusb.edu/ history/Mathematicians/Descartes.html (Descartes’ biography)
• www.saintjoe.edu/ karend/m122/CountingSlides.ppt (Sum and Product rules)
Problem 1.2.1. In a group of students, each person studies one and only one of three foreign languages: six people take French, eight take German, and nine students take Spanish. How many students are there in the group?

Solution. Denote the set of all students in the group by $X$, the subset of students studying French by $X_F$, the subset of students studying German by $X_G$, and the subset of students studying Spanish by $X_S$. Since each student studies at least one language, we can represent $X$ as the union,
$$X=X_F \cup X_G \cup X_S .$$

## 数学代写|组合学代写Combinatorics代考|The Sum Rule

If the finite sets $X_1, X_2, \ldots, X_m$ form a partition of a set $X$, then
$$|X|=\left|X_1\right|+\left|X_2\right|+\cdots+\left|X_m\right| .$$

Evidently, (1.2.1) is equivalent to Lemma 1.1.2.
If there is a sum rule, then there likely is a product rule. To introduce it, we again analyze a model problem.

Problem 1.2.2. Identification cards on Small Planet contain two characters, one capital Latin letter and one Hindu-Arabic digit, for example, ” $S-8$ “. How many various cards are there, if we can use all 26 letters and 10 digits?

Solution. First of all, we have to state unequivocally what cards must be considered identical, and what cards are different. Since we consider a mathematical problem, we do not take into consideration size, color, font, etc. Two cards are considered as different, if they have different pairs of symbols, that is, if at least one symbol on either card is distinct from the corresponding symbol on another card. In other words, to say that two cards are identical is just to say that they have both the same letter and the same digit. We reiterate here this statement, because clear qualification of what objects are distinct in a combinatorial problem and which ones are the same (are identical) is a crucial step in solving the problem; otherwise, two people can read the same words but solve two different problems.

## 数学代写|组合学代写Combinatorics代考|The sum and product rules

• www.math.csusb.edu/history/Mathematicians/Descartes.html (笛卡尔的传记)
• www.saintjoe.edu/karend/m122/CountingSlides.ppt (求和规则)
问题 1.2.1。在一组学生中，每个人学习三种外语中的一种，并且只学习一种: 六个人学习法 语，八个学习德语，九个学生学习西班牙语。小组中有多少学生?
解决方案。用以下表示组中所有学生的集合 $X$, 学习法语的学生子集 $X_F$, 学习德语的学生子集 $X_G$ ， 以及学习西班牙语的学生子集 $X_S$. 由于每个学生至少学习一种语言，我们可以代表 $X$ 作为工会，
$$X=X_F \cup X_G \cup X_S$$

## 数学代写|组合学代写Combinatorics代考|The Sum Rule

$$|X|=\left|X_1\right|+\left|X_2\right|+\cdots+\left|X_m\right| .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。