Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Fundamentals of Electrical Engineering I » Permutations and Combinations

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This collection is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange Grove

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Rice DSS - Braille display tagshide tags

    This collection is included inLens: Rice University Disability Support Services's Lens
    By: Rice University Disability Support Services

    Comments:

    "Electrical Engineering Digital Processing Systems in Braille."

    Click the "Rice DSS - Braille" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Rice Digital Scholarship display tagshide tags

    This collection is included in aLens by: Digital Scholarship at Rice University

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Bookshare

    This collection is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech Initiative

    Comments:

    "Accessible versions of this collection are available at Bookshare. DAISY and BRF provided."

    Click the "Bookshare" link to see all content affiliated with them.

  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "The course focuses on the creation, manipulation, transmission, and reception of information by electronic means. It covers elementary signal theory, time- and frequency-domain analysis, the […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Lens for Engineering

    This module and collection are included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Permutations and Combinations

Module by: Don Johnson. E-mail the author

Summary: Discusses the basics of combinations and permutations, and how to calculate the probability of certain events, such as n-bit errors in a codeword.

Permutations and Combinations

The lottery "game" consists of picking kk numbers from a pool of nn. For example, you select 66 numbers out of 6060. To win, the order in which you pick the numbers doesn't matter; you only have to choose the right set of 66 numbers. The chances of winning equal the number of different length-kk sequences that can be chosen. A related, but different, problem is selecting the batting lineup for a baseball team. Now the order matters, and many more choices are possible than when order does not matter.

Answering such questions occurs in many applications beyond games. In digital communications, for example, you might ask how many possible double-bit errors can occur in a codeword. Numbering the bit positions from 11 to NN, the answer is the same as the lottery problem with k=6k6. Solving these kind of problems amounts to understanding permutations - the number of ways of choosing things when order matters as in baseball lineups - and combinations - the number of ways of choosing things when order does not matter as in lotteries and bit errors.

Calculating permutations is the easiest. If we are to pick kk numbers from a pool of nn, we have nn choices for the first one. For the second choice, we have n1 n 1 . The number of length-two ordered sequences is therefore be n(n1) n n 1 . Continuing to choose until we make kk choices means the number of permutations is n(n1)(n2)(nk+1) n n 1 n 2 n k 1 . This result can be written in terms of factorials as n!(nk)! n n k , with n!=n(n1)(n2)1 n n n 1 n 2 1 . For mathematical convenience, we define 0!=1 0 1 .

When order does not matter, the number of combinations equals the number of permutations divided by the number of orderings. The number of ways a pool of kk things can be ordered equals k! k. Thus, once we choose the nine starters for our baseball game, we have 9!=362,880 9 362,880 different lineups! The symbol for the combination of kk things drawn from a pool of nn is nk n k and equals n!(nk)!k! n n k k .

Exercise 1

What are the chances of winning the lottery? Assume you pick 66 numbers from the numbers 11-6060.

Solution

606=60!54!6!=50,063,860 60 6 60 54 6 50,063,860 .

Combinatorials occur in interesting places. For example, Newton derived that the nn-th power of a sum obeyed the formula x+yn=n0xn+n1xn1y+n2xn2y2++nnyn x y n n 0 x n n 1 x n 1 y n 2 x n 2 y 2 n n y n .

Exercise 2

What does the sum of binomial coefficients equal? In other words, what is k=0nnk k 0 n n k

Solution

Because of Newton's binomial theorem, the sum equals 1+1n=2n 1 1 n 2 n .

A related problem is calculating the probability that any two bits are in error in a length-nn codeword when pp is the probability of any bit being in error. The probability of any particular two-bit error sequence is p21pn2 p 2 1 p n 2 . The probability of a two-bit error occurring anywhere equals this probability times the number of combinations: n2p21pn2 n 2 p 2 1 p n 2 . Note that the probability that zero or one or two, etc. errors occurring must be one; in other words, something must happen to the codeword! That means that we must have n01pn+n1p1pn1+n2p21pn2++nnpn=1 n 0 1 p n n 1 p 1 p n 1 n 2 p2 1 p n 2 n n p n 1 . Can you prove this?

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks