Factorial
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
(By now, factorials have gotten as big as 60 digits long!)
( is 65 digits long and has 12 terminal zeroes already!)
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)
See Also
- A cool link to calculate factorials: http://www.nitrxgen.net/factorialcalc.php