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[39] [40] The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials. [ 41 ] Factorials are used extensively in probability theory , for instance in the Poisson distribution [ 42 ] and in the probabilities of random permutations . [ 43 ]
The ordinary factorial, when extended to the gamma function, has a pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation!! = ()!! to give !! = (+)!! +.
The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in the OEIS). In this article, a factorial number representation will be flagged by a subscript "!". In addition, some examples will have digits delimited by a colon. For example, 3:4:1:0:1:0! stands for
When the variable is a positive integer, the number () is equal to the number of n-permutations from a set of x items, that is, the number of ways of choosing an ordered list of length n consisting of distinct elements drawn from a collection of size .
However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied.
One may show by induction that F(n) counts the number of ways that a n × 1 strip of squares may be covered by 2 × 1 and 1 × 1 tiles. On the other hand, if such a tiling uses exactly k of the 2 × 1 tiles, then it uses n − 2k of the 1 × 1 tiles, and so uses n − k tiles total.
There are finitely many natural numbers less than , so the number is guaranteed to reach a periodic point or a fixed point less than , making it a preperiodic point. For b = 2 {\displaystyle b=2} , the number of digits k ≤ n {\displaystyle k\leq n} for any number, once again, making it a preperiodic point.
n! (n factorial) is the number of n-permutations; !n (n subfactorial) is the number of derangements – n-permutations where all of the n elements change their initial places. In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position.