Design of experiments with full factorial design ( left ), response surface with second-degree polynomial ( right )

In mathematics, the gamma function ( represented by the capital Greek letter Γ ) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

In another study, three nullification instructions varying in explicitness as to nullification were combined with three criminal cases to yield a 3 × 3 factorial design.

This describes the factorial as a recursive function, with a single terminating base case.

The set of injective functions from X to Y may be denoted Y < sup >< u > X </ u ></ sup > using a notation derived from that used for falling factorial powers, since if X and Y are finite sets with respectively m and n elements, the number of injections from X to Y is n < sup >< u > m </ u ></ sup > ( see the twelvefold way ).

The same function can be expressed with clausal function definitions where the if-then-else conditional is replaced by a sequence of templates of the factorial function evaluated for specific values, separated by '|', which are tried one by one in the order written until a match is found:

Unlike other functions, the factorial function is denoted with the exclamation mark ( serving as the symbol of the function ) after the variable ( postfix notation ).

The factorial function, which defines the number of permutations on a set of fixed objects, grows very rapidly with the number of objects.

What exactly " programming at compile-time " means can be illustrated with an example of a factorial function, which in non-template C ++ can be written using recursion as follows:

With large numbers of attributes, the consideration task for respondents becomes too large and even with fractional factorial designs the number of profiles for evaluation can increase rapidly.

" denotes the factorial function ; compare this with Wilson's theorem, which states that every prime p divides ( p − 1 )!

* ( Pi function ) – the Gamma function when offset to coincide with the factorial

( 6 factorial ), a composite number with thirty divisors, more than any number below, making it a highly composite number.

This list has some items that would not fit in such a classification, such as list of exponential topics and list of factorial and binomial topics, which may surprise the reader with the diversity of their coverage.

Some industrialization of the periphery is possible under the condition of low wages, which, together with rising productivity, determine that unequal exchange sets in ( double factorial terms of trade < 1. 0 ; see Raffer, 1987 )

In probability theory, the nth factorial moment of a probability distribution, also called the nth factorial moment of any random variable X with that probability distribution, is

For example, if X has a Poisson distribution with expected value λ, then the nth factorial moment of X is

* ( Pi function ), the Gamma function when offset to coincide with the factorial

A classic example of recursion is computing the factorial, which is defined recursively as and To compute the value of the function on a given input, the function calls itself with a different input, and repeats until it terminates.

They can be explicitly related to the Stieltjes constants by re-expressing the falling factorial as a polynomial with Stirling number coefficients, and then solving.

with m given by the multiset coefficient, more familiarly the binomial coefficient, or more elegantly the rising factorial, as:

Designed experiments with full factorial design ( left ), response surface with second-degree polynomial ( right )

More examples of the use of the empty product in mathematics may be found in the binomial theorem, factorial, fundamental theorem of arithmetic, birthday paradox, Stirling number, König's theorem, binomial type, difference operator, Pochhammer symbol, proof that e is irrational, prime factor, binomial series, and multiset.

In fact, the more general concept of factorial was found at the same time by Arbogast.

For example, having decided to use < sub > q </ sub > as the q-analog of n, one may define the q-analog of the factorial, known as the q-factorial, by

A factorial prime is a prime number that is one less or one more than a factorial ( all factorials above 1 are even ).

* 119 is the smallest composite number that is 1 less than a factorial ( 120 is 5!

Also the largest number whose factorial is less than 10 < sup > 1000 </ sup >

The factorial number system is a mixed radix numeral system: the i-th digit from the right has base i, which means that the digit must be strictly less than i, and that ( taking into account the bases of the less significant digits ) its value to be multiplied by!

Functions included square root, inverse, trigonometric ( sine, cosine, tangent and their inverses ), exponentiation, logarithms and factorial.

levels of each factor, it is termed factorial.

Consequently, factorial designs are heavily used.

where Γ ( z ) is the gamma function, a generalization of the factorial function to non-integer values.

: Use of factorial experiments instead of the one-factor-at-a-time method.

That expansion, in turn, serves as the starting point for one of the derivations of precise error estimates for Stirling's approximation of the factorial function.

A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of n. The simple formula for the factorial, n < nowiki >!</ nowiki > = 1 × 2 × … × n, cannot be used directly for fractional values of n since it is only valid when n is a natural number ( i. e., a positive integer ).

Stirling's approximation is asymptotically equal to the factorial function for large values of n.

The Bohr – Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex, that is, its natural logarithm is convex on the positive real axis.

In a commercial setting, watering frequency is multi factorial and governed by computers or PLCs.

This implementation of the factorial function is not guaranteed to terminate, since a negative argument causes an infinite descending chain of recursive calls.

For example: addition, division, factorial, exponential and the nth prime are all primitive recursive.

( N factorial ) distinct terms.

A unary operator for which the reverse Polish notation is the general convention is the factorial.

In this situation 42 is called the message receiver, while ' factorial ' is the message selector.

" factorial " above is what is called a unary message because only one object, the receiver, is involved.

will change the meaning so that the expression first computes " 3 factorial + 4 " yielding 10.

That 10 then receives the second " factorial " message, yielding 3628800.

which sends " factorial " to 3, then " factorial " to the result ( 6 ), then " log " to the result ( 720 ), producing the result 2. 85733.