Sideway
output.to from Sideway
Draft for Information Only

Content

Algebra
 Permutations and Combinations
  Permutations
   Permutations of all at a time
   Proof
   Permutations of 𝑟 thing at a time
   Proof
  Combinations
   Combinations of 𝑟 thing at a time
  Homogeneous Products
   Proof
  Permutations of alike things
  Combinations of 𝑝 things found
  Theorem of Combination
  Combination of Different Things
 Sources and References

Algebra

Permutations and Combinations

Permutations

Permutations of all at a time

The number of permutations of 𝑛 things taken all at a time: =𝑛(𝑛−1)(𝑛−2)⋯3⋅2⋅1≡𝑛! or 𝑛(𝑛)

Proof

Proof by Induction. Assume the formula to be true for 𝑛 things. Now take 𝑛+1 things. After each of these the remaining 𝑛 things may be arranged in 𝑛! ways, making in all 𝑛×𝑛!, that is (𝑛+1)!, permutations of 𝑛+1 things; therefore, ⋯.

Permutations of 𝑟 thing at a time

The number of permutations of 𝑛 things taken 𝑟 at a time is denoted by 𝑃(𝑛,𝑟). 𝑃(𝑛,𝑟)=𝑛(𝑛)(𝑛−1)(𝑛−2)⋯(𝑛−𝑟+1)≡𝑛(𝑟).

Proof

Proof. By permutations of 𝑛 things; for (𝑛−𝑟) things are left out of each permutation; therefore 𝑃(𝑛,𝑟)=𝑛!÷(𝑛−𝑟)!. Observe that 𝑟=the number of factors.

Combinations

Combinations of 𝑟 thing at a time

The number of combinations of 𝑛 things taken 𝑟 at a time is denoted by 𝐶(𝑛,𝑟). 𝐶(𝑛,𝑟)=𝑛(𝑛)(𝑛−1)(𝑛−2)⋯(𝑛−𝑟+1)1⋅2⋅3⋯𝑟𝑛(𝑟)𝑟!  =𝑛!𝑟!(𝑛−𝑟)!=𝐶(𝑛,𝑛−𝑟) For every combination of 𝑟 things admits of 𝑟! permutations; therefore 𝐶(𝑛,𝑟)=𝑃(𝑛,𝑟)÷𝑟!. 𝐶(𝑛,𝑟) is greatest when 𝑟=12𝑛 or 12(𝑛±1) according as 𝑛 is even or odd.

Homogeneous Products

The number of homogeneous products of 𝑟 dimensions of 𝑛 things is denoted by 𝐻(𝑛,𝑟). 𝐻(𝑛,𝑟)=𝑛(𝑛)(𝑛+1)(𝑛+2)⋯(𝑛+𝑟−1)1⋅2⋅3⋯𝑟(𝑛+𝑟−1)(𝑟)𝑟! When 𝑟 is >, this reduces to (𝑟+1)(𝑟+2)⋯(𝑛+𝑟−1)(𝑛−1)!

Proof

𝐻(𝑛,𝑟) is equal to the number of terms in the product of the expansions by the Binomial Theorem of the 𝑛 expressions (1−𝑎𝑥)−1, (1−𝑏𝑥)−1, (1−𝑐𝑥)−1, ⋯. Put 𝑎=𝑏=𝑐=⋯=1. The number will be the coefficient of 𝑥𝑟 in (1−𝑥)−𝑛.

Permutations of alike things

The number of permutations of 𝑛 things taken all together, when 𝑎 of them are alike, 𝑏 of them alike, 𝑐 alike, ⋯. =𝑛!𝑎!𝑏!𝑐!⋯ For, if the 𝑎 things were all different, they would form 𝑎! permutations where there is now but one. so of 𝑏, 𝑐, ⋯.

Combinations of 𝑝 things found

The number of combinations of 𝑛 things 𝑟 at a time, in which any 𝑝 of them will always be found, is =𝐶(𝑛−𝑝,𝑟−𝑝) For, if the 𝑝 things be set on one side, we have to add to them 𝑟−𝑝 things taken from the remaining 𝑛−𝑝 things in every possible way.

Theorem of Combination

𝐶(𝑛−1,𝑟−1)+𝐶(𝑛−1,𝑟)=𝐶(𝑛,𝑟) Proof by induction or as follows: Put one out of 𝑛 letters aside; there are 𝐶(𝑛−1,𝑟) combinations of the remaining 𝑛−1 letters 𝑟 at a time. To complete the total 𝐶(𝑛,𝑟), we must place with the excluded letter all the combinations of the remaining 𝑛−1 letters 𝑟−1 at a time.

Combination of Different Things

If ther be one set of 𝑃 things, another of 𝑄 things, another of 𝑅 things, and so on; the number of combinations formed by taking one out of each set is =𝑃𝑄𝑅⋯, the product of the numbers in the several sets.
For one of the 𝑃 things will form 𝑄 combinations with the 𝑄 things. A second of the 𝑃 things will form 𝑄 more combinations; ans so on. In all, 𝑃𝑄 combinations of two things. Similarly there will be 𝑃𝑄𝑅 combinations of three things; and so on.
On the same principle, if 𝑝, 𝑞, 𝑟, ⋯ things be taken out of each set respectively, the number of combinations will be the product of the numbers of the separate combinations; that is, =𝐶(𝑃𝑝)⋅𝐶(𝑄𝑞)⋅𝐶(𝑅𝑟)⋅⋯ The number of combinations of 𝑛 things taken 𝑚 at a time, when 𝑝 of the 𝑛 things are alike, 𝑞 of them alike, 𝑟 of them alike, ⋯, will be the sum of all the combinations of each possible form of 𝑚 dimensions, and this is equal to the coefficient of 𝑥𝑚 in the expansion of (1+𝑥+𝑥2+⋯+𝑥𝑝)(1+𝑥+𝑥2+⋯+𝑥𝑞)(1+𝑥+𝑥2+⋯+𝑥𝑟)⋯ The total number of possible combinations under the same circumstances, when the 𝑛 things are taken in all ways, 1, 2, 3, ⋯, 𝑛 at a time, =(𝑝+1)(𝑞+1)(𝑟+1)⋯−1 The number of permutations when they are taken 𝑚 at a time in all possible ways will be equal to the product of 𝑚! and the coefficient of 𝑥𝑚 in the expansion of 1+𝑥+𝑥22!+𝑥33!+⋯+𝑥𝑝𝑝!1+𝑥+𝑥22!+𝑥33!+⋯+𝑥𝑞𝑞!

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive

©sideway

ID: 210600008 Last Updated: 6/8/2021 Revision: 0 Ref:

close

References

  1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
  2. Wheatstone, C., 1854, On the Formation of Powers from Arithmetical Progressions
  3. Stroud, K.A., 2001, Engineering Mathematics
  4. Coolidge, J.L., 1949, The Story of The Binomial Theorem
close
IMAGE

Home 5

Business

Management

HBR 3

Information

Recreation

Hobbies 8

Culture

Chinese 1097

English 337

Reference 68

Computer

Hardware 154

Software

Application 207

Digitization 25

Latex 35

Manim 203

Numeric 19

Programming

Web 285

Unicode 504

HTML 65

CSS 63

SVG 8

ASP.NET 211

OS 422

DeskTop 7

Python 64

Knowledge

Mathematics

Formulas 8

Algebra 84

Number Theory 206

Trigonometry 31

Geometry 32

Coordinate Geometry 1

Calculus 67

Complex Analysis 21

Engineering

Tables 8

Mechanical

Mechanics 1

Rigid Bodies

Statics 92

Dynamics 37

Fluid 5

Fluid Kinematics 5

Control

Process Control 1

Acoustics 19

FiniteElement 2

Natural Sciences

Matter 1

Electric 27

Biology 1

Geography 1


Copyright © 2000-2021 Sideway . All rights reserved Disclaimers last modified on 06 September 2019