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The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast, [18] in the first work on Faà di Bruno's formula, [19] but referring to a more general concept of products of arithmetic progressions. The "factors" that this name refers to are the terms of the product formula for the factorial. [20]
(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.
The ratio of the factorial!, that counts all permutations of an ordered set S with cardinality, and the subfactorial (a.k.a. the derangement function) !, which counts the amount of permutations where no element appears in its original position, tends to as grows.
De Polignac's formula; Difference operator; Difference polynomials; Digamma function; Egorychev method; Erdős–Ko–Rado theorem; Euler–Mascheroni constant; Faà di Bruno's formula; Factorial; Factorial moment; Factorial number system; Factorial prime; Gamma distribution; Gamma function; Gaussian binomial coefficient; Gould's sequence ...
In this formula and in many other places, the falling factorial () in the calculus of finite differences plays the role of in differential calculus. Note for instance the similarity of Δ ( x ) n = n ( x ) n − 1 {\displaystyle \Delta (x)_{n}=n(x)_{n-1}} to d d x x n = n x n − 1 {\displaystyle {\frac {\textrm {d}}{{\textrm {d}}x}}x^{n}=nx^{n ...
Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .
A more efficient method to compute individual binomial coefficients is given by the formula = _! = () (()) () = = +, where the numerator of the first fraction, _, is a falling factorial. This formula is easiest to understand for the combinatorial interpretation of binomial coefficients.
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre . It is also sometimes known as de Polignac's formula , after Alphonse de Polignac .