Difference between revisions of "Factorial"
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== Examples == | == Examples == | ||
− | * <math>0! = 1</math> (remember! this is 1, not 0!) | + | * <math>0! = 1</math> (remember! this is 1, not 0! (the '!' was an exclamation mark, not a factorial sign)) |
* <math>1! = 1</math> | * <math>1! = 1</math> | ||
* <math>2! = 2</math> | * <math>2! = 2</math> | ||
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==Problems== | ==Problems== | ||
===Introductory=== | ===Introductory=== | ||
− | *Find the | + | *Find the unitsdigit of the sum |
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath> | <cmath>\sum_{i=1}^{100}(i!)^{2}</cmath> | ||
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<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math> | <math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math> | ||
([[2007 iTest Problems/Problem 6|Source]]) | ([[2007 iTest Problems/Problem 6|Source]]) | ||
+ | |||
===Intermediate=== | ===Intermediate=== | ||
*<math>\frac{((3!)!)!}{3!}=k*n!</math>, where <math>k</math> and <math>n</math> are positive integers and <math>n</math> is as large as possible. Find the value of <math>k+n</math>. | *<math>\frac{((3!)!)!}{3!}=k*n!</math>, where <math>k</math> and <math>n</math> are positive integers and <math>n</math> is as large as possible. Find the value of <math>k+n</math>. | ||
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* A cool link to calculate factorials: http://www.nitrxgen.net/factorialcalc.php | * A cool link to calculate factorials: http://www.nitrxgen.net/factorialcalc.php | ||
− | On that link, you can calculate factorials from <math>0!</math> to as much as <math> | + | On that link, you can calculate factorials from <math>0!</math> to as much as <math>100000!</math> |
Latest revision as of 15:40, 17 March 2024
The factorial is an important function in combinatorics and analysis, used to determine the number of ways to arrange objects.
Contents
[hide]Factorials Video
Definition
The factorial is defined for positive integers as . Alternatively, a recursive definition for the factorial is .
Examples
- (remember! this is 1, not 0! (the '!' was an exclamation mark, not a factorial sign))
- (Note: this number is 82 digits long with 14 terminal zeroes!)
- (Note: This number is 2568 digits long and has as much as 249 terminal zeroes!)
- is 38660 digits long and has 2499 terminal zeroes!
- is 456574 digits long and has 24999 terminal zeroes!
- is 973751 digits long and has 49998 terminal zeroes!
Additional Information
By convention and rules of an empty product, 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 unitsdigit of the sum
(Source)
Intermediate
- , where and are positive integers and is as large as possible. Find the value of .
(Source)
- 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
On that link, you can calculate factorials from to as much as