Factorial
The factorial is an important function in combinatorics and analysis, used to determine the number of ways to arrange objects.
Contents
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
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