Difference between revisions of "Factorial"
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The factorial is defined for [[positive integer]]s as <math>n!=n \cdot (n-1) \cdots 2 \cdot 1 = \prod_{i=1}^n i</math>. Alternatively, a [[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>. | The factorial is defined for [[positive integer]]s as <math>n!=n \cdot (n-1) \cdots 2 \cdot 1 = \prod_{i=1}^n i</math>. Alternatively, a [[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>. | ||
+ | |||
+ | == Examples == | ||
+ | * <math>0! = 1</math> | ||
+ | * <math>1! = 1</math> | ||
+ | * <math>2! = 2</math> | ||
+ | * <math>3! = 6</math> | ||
+ | * <math>4! = 24</math> | ||
+ | * <math>5! = 120</math> | ||
+ | * <math>6! = 720</math> | ||
+ | * <math>7! = 5040</math> | ||
+ | * <math>8! = 40320</math> | ||
+ | * <math>9! = 362880</math> | ||
+ | * <math>10! = 3628800</math> | ||
+ | * <math>11! = 39916800</math> | ||
+ | * <math>12! = 479001600</math> | ||
+ | * <math>13! = 6227020800</math> | ||
+ | * <math>14! = 87178291200</math> | ||
+ | * <math>15! = 1307674368000</math> | ||
+ | * <math>16! = 20922789888000</math> | ||
+ | * <math>17! = 355687428096000</math> | ||
+ | * <math>18! = 6402373705728000</math> | ||
+ | * <math>19! = 121645100408832000</math> | ||
+ | * <math>20! = 2432902008176640000</math> | ||
== Additional Information == | == Additional Information == |
Revision as of 20:45, 29 March 2011
The factorial is an important function in combinatorics and analysis, used to determine the number of ways to arrange objects.
Contents
[hide]Definition
The factorial is defined for positive integers as . Alternatively, a recursive definition for the factorial is
.
Examples
Additional Information
By convention, is given the value
.
The gamma function is a generalization of the factorial to values other than nonnegative integers.
Prime Factorization
- Main article: Prime factorization
Documentation
Use {{hatnote|text}} </noinclude>
Since is the product of all positive integers not exceeding
, it is clear that it is divisible by all
primes
, and not divisible by any prime
. But what is the power of a prime
in the prime factorization of
? We can find it as the sum of powers of
in all the factors
;
but rather than counting the power of
in each factor, we shall count the number of factors divisible by a given power of
. Among the numbers
, exactly
are divisible by
(here
is the floor function). The ones divisible by
give one power of
. The ones divisible by
give another power of
. Those divisible by
give yet another power of
. Continuing in this manner gives
for the power of in the prime factorization of
. The series is formally infinite, but the terms converge to
rapidly, as it is the reciprocal of an exponential function. For example, the power of
in
is just
(
is already greater than
).
Uses
The factorial is used in the definitions of combinations and permutations, as is the number of ways to order
distinct objects.
Problems
Introductory
- Find the units digit of the sum
(Source)
Intermediate
- Let
be the product of the first
positive odd integers. Find the largest integer
such that
is divisible by
(Source)
Olympiad
- Let
be the number of permutations of the set
, which have exactly
fixed points. Prove that
.
(Source)