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In mathematics, the factorial of a non-negative integer, denoted by !, is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: ! = () = ()! For example, ! =! = =
The combinatorial interpretation of b 0 is the number of 0-tuples of elements from a b-element set; there is exactly one 0-tuple. The set-theoretic interpretation of b 0 is the number of functions from the empty set to a b-element set; there is exactly one such function, namely, the empty function. [1] All three of these specialize to give 0 0 = 1.
(The place value is the factorial of one less than the radix position, which is why the equation begins with 5! for a 6-digit factoradic number.) ... 0:1:0!). In fact ...
For example, the empty products 0! = 1 (the factorial of zero) and x 0 = 1 shorten Taylor series notation (see zero to the power of zero for a discussion of when x = 0). Likewise, if M is an n × n matrix, then M 0 is the n × n identity matrix , reflecting the fact that applying a linear map zero times has the same effect as applying the ...
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
One way of stating the approximation involves the logarithm of the factorial: (!) = + (), where the big O notation means that, for all sufficiently large values of , the difference between (!
Let be a natural number. For a base >, we define the sum of the factorials of the digits [5] [6] of , :, to be the following: = =!. where = ⌊ ⌋ + is the number of digits in the number in base , ! is the factorial of and
In this article, the symbol () is used to represent the falling factorial, and the symbol () is used for the rising factorial. These conventions are used in combinatorics , [ 4 ] although Knuth 's underline and overline notations x n _ {\displaystyle x^{\underline {n}}} and x n ¯ {\displaystyle x^{\overline {n}}} are increasingly popular.