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:<big><math>0! = (1-1)! = \frac{1!}{1} = \frac{1}{1} = 1</math></big>
:<big><math>0! = (1-1)! = \frac{1!}{1} = \frac{1}{1} = 1</math></big>
==Permutations==
Factorials frequently make an appearance in the mathematics of ''combinatorics'' and statistics. The most direct application concerns the number of ''permutations'' of a set of distinguishable objects—that is, the number of ways n different things can be arranged.
In general, n different things can be arranged in n! ways. For example, three objects can be arranged in 3! = 6 different ways:
:A B C
:A C B
:B A C
:B C A
:C A B
:C B A
It is easy to see that this is always true. We have seen that three objects can be arranged in six ways. If we now add a fourth object, D, we can see that for each arrangement there are four different places where we can insert the D. For example, starting with A B C, we can get
:'''D''' A B C
:A '''D''' B C
:A B '''D''' C
:A B C '''D'''
Corresponding to each permutation of three objects, there are four permutations of four objects, so there are 4 * 6 = 24 permutations of four objects. Similarly, there are 5 * 24 = 120 permutations of five, 6 * 120 = 720 permutations of six, and so forth.
==Change-ringing==
'''Change-ringing''' has been a popular hobby or pastime in England since the 1600s. It consists of ringing a set of bells, such as the bells in a church steeple, with the goal of ringing them every possible permutation. This is called a "full peal."
[[Dorothy Sayers]] wrote a mystery novel entitled [[The Nine Tailors]] in which change-ringing forms an important part of the story.
==The number 7! in Plato's ''Laws''==
==The number 7! in Plato's ''Laws''==