So far combinations and permutations are discussed with different theorems in algebra. In this paper I apply assembly analysis to get the theorems easy and memorable. After assembly analysis applied there becomes a lot of new theorems and all the theorems get a new face by summation methods. Formations mean the selections of a random experiment where order is not taken into account and repartitions are allowed. Homogenations mean the selections of a random experiment where order is taken into account and repartitions are allowed.
Table of Contents
1. Introduction
2. Findings
2.1 Combination space
2.2 Combination event
2.3 General combination theorem
2.4 Combination distribution
2.4.1 Moments
2.5 Permutation space
2.6 Permutation event
2.7 General permutation theorem
2.8 Permutation distribution
2.8.1 Moments
2.9 Formation space
2.10 Formation event
2.11 General formation theorem
2.12 Formation distribution
2.12.1 Moments
2.13 Homogenation space
2.14 Homogenation event
2.15 Multinomial expression
2.16 General homogenation theorem
2.17 Homogenation distribution
3. Conclusions
Objectives and Thematic Focus
This paper aims to simplify and unify algebraic theorems regarding combinations and permutations by applying assembly analysis and summation methods, while introducing new concepts known as formations and homogenations. The research seeks to redefine these mathematical outcomes through a structured framework where component order and repartition possibilities are explicitly distinguished.
- Application of assembly analysis to algebraic theorems.
- Development of summation methods for combinatorial and permutational calculations.
- Introduction and formalization of "Formation" and "Homogenation" as random experiment outcomes.
- Definition of distribution functions and moments for all four member types (combinations, permutations, formations, homogenations).
- Establishment of general theorems for identifying and calculating specific component configurations.
Excerpt from the Book
2.1 Combination space
A combination space is a set of all possible combinations (outcomes) of an experiment from a parent assembly A where the outcomes do not take order of the components into account. Let a combination space contains T possible outcomes then the combination space denoted by C{A/V} is
C{A/V} = {C1, C2, C3, ..., Ct, ..., CT}
where, V = 1, 2, 3, ........., N
N = Parent component number.
Example 1: Set a combination space of the experiment “4 letters a, b, c, and d select 3 at a time ”.
Solution: We have given A = (a, b, c, d) and V = 3.
Thus the combination space is
C{(a, b, c, d)/3} = {(a, b, c), (a, b, d ), (a, c, d ), (b, c, d)}.
Chapter Summary
1. Introduction: Defines the core concepts of combinations, permutations, formations, and homogenations, highlighting the role of component designation and order.
2. Findings: Provides the comprehensive mathematical derivation of spaces, events, and distribution theorems for combinations, permutations, formations, and homogenations.
2.1 Combination space: Defines the set of outcomes where component order is irrelevant, illustrated with a selection example.
2.2 Combination event: Details subsets of combination spaces with specific identified components.
2.3 General combination theorem: Establishes a theorem for combinations when selecting specific sub-components from a limited set.
2.4 Combination distribution: Introduces the probability mass function for combination variates.
2.5 Permutation space: Defines outcomes where component order is taken into account.
2.6 Permutation event: Defines subsets of permutation spaces with fixed initial components.
2.7 General permutation theorem: Outlines the calculation for permutations with limited component sizes.
2.8 Permutation distribution: Presents the probability mass function for permutation variates.
2.9 Formation space: Introduces the concept of formations where order is ignored but repetitions are allowed.
2.10 Formation event: Describes subsets of formation spaces with identified components.
2.11 General formation theorem: Provides the calculation method for formations involving limited subsets of components.
2.12 Formation distribution: Defines the discrete distribution for formation variates.
2.13 Homogenation space: Defines the outcome space where order is significant and repetitions are allowed.
2.14 Homogenation event: Details subsets within homogenation spaces based on identified components.
2.15 Multinomial expression: Defines a specific algebraic expression used to solve homogenation theorems.
2.16 General homogenation theorem: Establishes the theorem for homogenations using the multinomial expression.
2.17 Homogenation distribution: Finalizes the mathematical treatment with the distribution of homogenation variates.
3. Conclusions: Summarizes the study's contribution in introducing "formations" and "homogenations" as novel mathematical concepts alongside traditional theory.
Keywords
Combinations, general combination theorem, combination distribution, permutations, general permutation theorem, permutation distribution, formations, general formation theorem, formation distribution, homogenations, equigenous expression, general homogenation theorem, homogenation distribution, assembly analysis, summation methods.
Frequently Asked Questions
What is the fundamental focus of this research paper?
The paper focuses on re-evaluating and expanding algebraic theorems for combinations and permutations through "assembly analysis" and introducing two new concepts: formations and homogenations.
What are the central thematic fields discussed?
The central fields are combinatorial algebra, probability distributions of discrete variables, and the classification of random experiment outcomes based on order and repetition.
What is the primary goal or research question?
The primary goal is to make algebraic theorems "easy and memorable" by applying new assembly analysis and summation methods, while formalizing the mathematical behavior of formations and homogenations.
Which scientific methods are utilized?
The author utilizes assembly analysis, summation methods, and the derivation of probability mass functions (multinomial expressions) to standardize the treatment of these four "Biswas members".
What is covered in the main body of the paper?
The main body systematically derives the spaces, events, general theorems, and probability distributions for all four outcome types: combinations, permutations, formations, and homogenations.
Which keywords characterize the work?
Key terms include combinations, permutations, formations, homogenations, assembly analysis, summation methods, and distribution variates.
How do formations differ from combinations?
In formations, components can be designated by die sides (repetitions allowed) and order is not considered, whereas in combinations, components cannot be designated this way.
How are homogenations distinguished from permutations?
While both consider order, homogenations specifically allow components to be designated by die sides (implying potential repetitions), unlike standard permutations.
What role does the "Multinomial Expression" play in the paper?
It is an algebraic tool introduced in section 2.15, essential for proving and calculating values within the general homogenation theorems.
- Citar trabajo
- Deapon Biswas (Autor), 2018, The Four Biswas Members, Múnich, GRIN Verlag, https://www.grin.com/document/450780
