### 数学代写|组合学代写Combinatorics代考|CS586

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合学代写Combinatorics代考|What is Combinatorics

Arithmetic studies the properties of natural numbers and the principles of manipulating them, known as the arithmetic operations (addition, subtraction, multiplication, and division). Plane geometry (planimetric) provides an interpretation of important patterns concerning such shapes as triangles, circles, trapezia, parallelograms, etc. In addition, what does combinatorics deal with? Probably the best way to form the correct vision of the subject of combinatorics is through the consideration of specific examples from its domain.
Example 1.1. Is there a way to place the numbers $1,2,3,4,5,6,7,8,9$ in a $3 \times 3$ square grid so that the sums of numbers in all rows, columns and diagonals are equal to the same value?

Clearly, this is not a complex problem. After several efforts, one almost inevitably reaches the desired placing. For example, the following:
$$\begin{array}{lll} 4 & 9 & 2 \ 3 & 5 & 7 \ 8 & 1 & 6 \end{array}$$
Hence, the answer to the question is positive. Moreover, it yields another one, much less trivial question: how many such $3 \times 3$ square grids exist?

Example 1.2. Let us have a drawing with small circles denoting cities and lines denoting routes between them. Departing from city $A$, is it possible to return to city $A$ by traveling each route exactly once (cities may be revisited more than once)? The answer is positive for the provided scheme of routes. Moreover, this is true for any city in the drawing. Explain the reasoning behind that fact. Which special feature (or features) should a scheme obtain in order for the answer to remain positive? Come up with the easiest possible scheme, which does not allow a journey with stated conditions.

## 数学代写|组合学代写Combinatorics代考|Combinatorial Rule of Product

Behind this solid name, there is simple content, and the simplicity hides pitfalls which a beginner utterly needs to learn to bypass.

Example 1.11. John eats in a café every day and every time follows the same rule: his meal consists of one entrée and one main course. There is a choice of five entrées and seven main courses today. How many options are there for John to configure his meal?

The problem can be stated in a different way changing the emphasis in the question. Assume there are always the same five entrees and seven main courses on the menu. How many days can pass with John choosing a new combination for his meal?

There is no doubt the reader has already found an answer. However, taking into account that the situation in the problem may arise in different variations in the future, and the necessity to recognize it in more complex cases, we outline the details of the explanation of the answer.

Let us adhere to the second formulation of the question. Assume John decided to use the following algorithm. He is going to choose the same entrée adding variability to his meals by the choice of the main course. How many days John can choose meals without repetition? Obviously, the answer is seven. On the eighth day, he has to change the entrée. John will have another seven days of different meals with this choice of entrée. The same will happen for the other three choices of entrée. Hence, having five entrees and seven main courses he can choose $5 \cdot 7=35$ different meals.
Example 1.12. How many two-digit numbers comprise odd digits only?
The answer to the question can be illustrated by Fig. 1.4. The first row and the first column of the table consists of all five odd numbers each. Consider the square circled by the double line. Every cell can be specified by two numbers: first, the one placed to the left from it, and then, the one above it. Putting these numbers next to each other, we get a two-digit number, which can be taken as a code of the corresponding cell. Thus, every cell has a code attached to it, and every code denotes a specific cell. For example, the crosshatched cell has code 57. The cells are geometric analogs of their codes, which are two-digit numbers comprising of odd digits. Hence, the numbers of codes (two-digit numbers) and cells (their geometric analogs) are equal. The latter amounts to $5 \cdot 5=25$ (five rows with five cells in each). Therefore, there is the same amount of two-digit numbers, which is the answer to the problem.

## 数学代写|组合学代写Combinatorics代考|What is Combinatorics

$$\begin{array}{lllllll} 4 & 9 & 23 & 5 & 78 & 1 & 6 \end{array}$$

## 有限元方法代写

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## 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。