Difference between revisions of "Factorial"
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* <math>29! = 8841761993739701954543616000000</math> | * <math>29! = 8841761993739701954543616000000</math> | ||
* <math>30! = 265252859812191058636308480000000</math> | * <math>30! = 265252859812191058636308480000000</math> | ||
+ | * <math>31! = 8222838654177922817725562880000000</math> | ||
+ | * <math>32! = 263130836933693530167218012160000000</math> | ||
+ | * <math>33! = 8683317618811886495518194401280000000</math> | ||
+ | * <math>34! = 295232799039604140847618609643520000000</math> | ||
+ | * <math>35! = 10333147966386144929666651337523200000000</math> | ||
+ | * <math>36! = 371993326789901217467999448150835200000000</math> | ||
+ | * <math>37! = 13763753091226345046315979581580902400000000</math> | ||
+ | * <math>38! = 523022617466601111760007224100074291200000000</math> | ||
+ | * <math>39! = 20397882081197443358640281739902897356800000000</math> | ||
+ | * <math>40! = 815915283247897734345611269596115894272000000000</math> | ||
+ | * <math>41! = 33452526613163807108170062053440751665152000000000</math> | ||
+ | * <math>42! = 1405006117752879898543142606244511569936384000000000</math> | ||
+ | * <math>43! = 60415263063373835637355132068513997507264512000000000</math> | ||
+ | * <math>44! = 2658271574788448768043625811014615890319638528000000000</math> | ||
+ | * <math>45! = 119622220865480194561963161495657715064383733760000000000</math> | ||
+ | * <math>46! = 5502622159812088949850305428800254892961651752960000000000</math> | ||
+ | * <math>47! = 258623241511168180642964355153611979969197632389120000000000</math> | ||
+ | (By now, factorials have gotten as big as 60 digits long!) | ||
+ | * <math>48! = 12413915592536072670862289047373375038521486354677760000000000</math> | ||
+ | * <math>49! = 608281864034267560872252163321295376887552831379210240000000000</math> | ||
+ | * <math>50! = 30414093201713378043612608166064768844377641568960512000000000000</math> | ||
+ | (<math>50!</math> is 65 digits long and has 12 terminal zeroes already!) | ||
== Additional Information == | == Additional Information == |
Revision as of 21:05, 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
(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)