Pronunciation: /fækˈtoʊr i əl/ ?

A factorial of n is the product of all integers from 1 to n, inclusive. The exclamation mark (!) is the unary operator that represents factorial. The expression '5!' is read, "Five factorial" and means 1·2·3·4·5 which equals 120. Zero factorial (0!) is treated as a special case and is defined to be 0! = 1. At this level of mathematics, the n in n! must be a non-negative integer.

In advanced mathematics, factorial is defined using the gamma function:

z! is defined as gamma(z+1) which is defined as the integral from 0 to infinity of e^(-t)*t^z dt.
Equation 1

First 11 Factorials

Table of Factorials
11! = 11
22! = 1·22
33! = 1·2·36
44! = 1·2·3·424
55! = 1·2·3·4·5120
66! = 1·2·3·4·5·6720
77! = 1·2·3·4·5·6·75040
88! = 1·2·3·4·5·6·7·840,320
99! = 1·2·3·4·5·6·7·8·9362,880
1010! = 1·2·3·4·5·6·7·8·9·103,628,800
Table 1

Working With Factorials

Since factorials get very large very quickly (see table 1), they can be difficult to calculate without a calculator. However, when used in mathematics, equations using factorials usually involve ratios of factorials. Since factorials by definition have many common factors, these problems can be reduced in difficulty by canceling common factors.

To understand this, start with the definition of a combination:

Combination of n objects taken k at a time equals n!/(k!(n-k)!).
Equation 1
If one wishes to calculate Combination of 8 objects taken 5 at a time., the formula is: Combination of 8 objects taken 5 at a time = 8!/(5!(8-5)!)..
First, write out the factorials: 8!/(5!3!)=1*2*3*4*5*6*7*8/(1*2*3*4*5*1*2*3).
Now use the property of multiplying by 1 to cancel all ones: 1*2*3*4*5*6*7*8/(1*2*3*4*5*1*2*3)=2*3*4*5*6*7*8/(2*3*4*5*2*3).
Now cancel the factors that are the same: 2*3*4*5*6*7*8/(2*3*4*5*2*3)=6*7*8/(2*3).
Now look for remaining factors that have common factors. In the case, 2·3 = 6 so the 6 on the top cancels the 2·3 on the bottom: 6*7*8/(2*3)=7*8/1=7*8.
This has now become a simple multiplication problem. Since 7*8 = 56, there are exactly 56 ways that five objects selected from a pool of eight objects can be arranged. So, Combination of 8 objects taken 5 at a time equals 56..


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  2. Bettinger, Alvin K. and Englund John A.. Algebra And Trigonometry, pg 97. International Textbook Company, January 1963. (Accessed: 2010-02-04). http://www.archive.org/stream/algebraandtrigon033520mbp#page/n114/mode/1up/search/factorial.
  3. Aley, Robert J. and Rothrock, David A.. The Essentials of Algebra for Secondary Schools, pg 272. Silver, Burdett and Company, 1904. (Accessed: 2010-02-04). http://www.archive.org/stream/cu31924031286143#page/n283/mode/1up/search/factorial.
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Factorial. 2010-02-04. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/factorial.html.


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2010-02-04: Added "References" (McAdams, David.)
2008-12-29: Corrected equations (McAdams, David.)
2008-04-23: Initial version (McAdams, David.)

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