Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » ENGR 2113 ECE Math » Probability Concepts -- 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.
  • Bookshare

    This module is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech InitiativeAs a part of collection: "Advanced Algebra II: Conceptual Explanations"

    Comments:

    "DAISY and BRF versions of this collection are available."

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

  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection: "Advanced Algebra II: Conceptual Explanations"

    Comments:

    "This is the "concepts" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about […]"

    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

  • Busbee's Math Materials display tagshide tags

    This module is included inLens: Busbee's Math Materials Lens
    By: Kenneth Leroy BusbeeAs a part of collection: "Advanced Algebra II: Conceptual Explanations"

    Click the "Busbee's Math Materials" link to see all content selected in this lens.

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

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.
 

Probability Concepts -- Combinations

Module by: Kenny M. Felder. E-mail the author

Summary: An introduction to combinations.

Let’s start once again with a deck of 52 cards. But this time, let’s deal out a poker hand (5 cards). How many possible poker hands are there?

At first glance, this seems like a minor variation on the Solitaire question above. The only real difference is that there are five cards instead of six. But in face, there is a more important difference: order does not matter. We do not want to count “Ace-King-Queen-Jack-Ten of spades” and “Ten-Jack-Queen-King-Ace of spades” separately; they are the same poker hand.

To approach such question, we begin with the permutations question: how many possible poker hands are there, if order does matter? 52 × 51 × 50 × 49 × 48 52×51×50×49×48, or 52!47!52!47! size 12{ { {"52"!} over {"47"!} } } {}. But we know that we are counting every possible hand many different times in this calculation. How many times?

The key insight is that this second question—“How many different times are we counting, for instance, Ace-King-Queen-Jack-Ten of spades?”—is itself a permutations question! It is the same as the question “How many different ways can these five cards be rearranged in a hand?” There are five possibilities for the first card; for each of these, four for the second; and so on. The answer is 5! which is 120. So, since we have counted every possible hand 120 times, we divide our earlier result by 120 to find that there are 52!(47!)(5!)52!(47!)(5!) size 12{ { {"52"!} over { \( "47"! \) \( 5! \) } } } {}, or about 2.6 Million possibilities.

This question—“how many different 5-card hands can be made from 52 cards?”—turns out to have a surprisingly large number of applications. Consider the following questions:

  • A school offers 50 classes. Each student must choose 6 of them to fill out a schedule. How many possible schedules can be made?
  • A basketball team has 12 players, but only 5 will start. How many possible starting teams can they field?
  • Your computer contains 300 videos, but you can only fit 10 of them on your iPod. How many possible ways can you load your iPod?

Each of these is a combinations question, and can be answered exactly like our card scenario. Because this type of question comes up in so many different contexts, it is given a special name and symbol. The last question would be referred to as “300 choose 10” and written 3001030010 size 12{ left ( matrix { "300" {} ## "10" } right )} {}. It is calculated, of course, as 300!(290!)(10!)300!(290!)(10!) size 12{ { {"300"!} over { \( "290"! \) \( "10"! \) } } } {} for reasons explained above.

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