Combination

Pronunciation: /ˌkɒm.bəˈneɪ.ʃən/ Explain

A combination is all possible combinations of k elements of a set of n objects without considering order.^[2][3] If one is not considering order, the combination ABC is the same as the combination ACB. The mathematical notation for combination is

This is read saying, "n objects taken k at a time".

The formula for calculating combinations is

The symbol ! means factorial. Factorial is defined as n! = 1 · 2 · 3 · … · n.

A simple example is

One can verify this using a set of two objects, for example {a, b}. How many ways can one of these two objects be selected? There are two ways: {a} and {b}.

Another example is

To verify this, start with a set of five objects: . How many ways can these be listed without regard to order?

References

1. McAdams, David E.. All Math Words Dictionary, combination. 2nd Classroom edition 20150108-4799968. pg 36. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Grinstead, Charles M. and Snell, J. Laurie. Introduction to Probability. pp 92-119. Last Accessed 6/25/2018. http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/amsbook.mac.pdf. Buy the book
3. Smith, Charles. A treatise on algebra. pp 282-288. www.archive.org. Macmillan and Company. 1892. Last Accessed 6/25/2018. http://www.archive.org/stream/cu31924060184433#page/n307/mode/2up/search/combination. Buy the book

More Information

• McAdams, David E.. Factorial. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018. http://www.allmathwords.org/en/f/factorial.html.

Cite this article as:

McAdams, David E. Combination. 4/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/combination.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
1/14/2010: Added "References". (McAdams, David E.)
1/12/2010: Added "References". (McAdams, David E.)
11/27/2008: Changed equations to images. Added Dictionary.com to 'More Information' (McAdams, David E.)
7/7/2008: Changed equations from images to Latex. (McAdams, David E.)
4/24/2008: Initial version. (McAdams, David E.)