Factorial/Examples

From ProofWiki
Jump to navigation Jump to search

Examples of Factorials

The factorials of the first few positive integers are as follows:

$\begin{array}{r|r} n & n! \\ \hline 0 & 1 \\ 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 24 \\ 5 & 120 \\ 6 & 720 \\ 7 & 5 \, 040 \\ 8 & 40 \, 320 \\ 9 & 362 \, 880 \\ 10 & 3 \, 628 \, 800 \\ \end{array}$

This sequence is A000142 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Factorial of $0$

The factorial of $0$ is $1$:

$0! = 1$


Factorial of $1$

The factorial of $1$ is $1$:

$1! = 1$


Factorial of $4$

$4! = 24$


Factorial of $5$

$5! = 120$


Factorial of $6$

$6! = 720$


Factorial of $10$

$10! = 3 \, 628 \, 000$


Factorial of $11$

$11! = 39 \, 916 \, 800$


Factorial of $12$

$12! = 479 \, 001 \, 600$


Factorial of $13$

$13! = 6 \, 227 \, 020 \, 800$


Factorial of $14$

$14! = 87 \, 178 \, 291 \, 200$


Factorial of $15$

$15! = 1 \, 307 \, 674 \, 368 \, 000$


Factorial of $16$

$16! = 20 \, 922 \, 789 \, 888 \, 000$


Factorial of $17$

$17! = 355 \, 687 \, 428 \, 096 \, 000$


Factorial of $18$

$18! = 6 \, 402 \, 373 \, 705 \, 728 \, 000$


Factorial of $19$

$19! = 121 \, 645 \, 100 \, 408 \, 832 \, 000$


Factorial of $20$

$20! = 2 \, 432 \, 902 \, 008 \, 176 \, 640 \, 000$


Factorial of $21$

$21! = 51 \, 090 \, 942 \, 171 \, 709 \, 440 \, 000$


Factorial of $22$

$22! = 1 \, 124 \, 000 \, 727 \, 777 \, 607 \, 680 \, 000$


Factorial of $23$

$23! = 25 \, 852 \, 016 \, 738 \, 884 \, 976 \, 640 \, 000$


Factorial of $24$

$24! = 620 \, 448 \, 401 \, 733 \, 239 \, 439 \, 360 \, 000$


Factorial of $25$

$25! = 15 \, 511 \, 210 \, 043 \, 330 \, 985 \, 984 \, 000 \, 000$


Prime Factors of $39!$

The prime decomposition of $39!$ is given as:

$39! = 2^{35} \times 3^{18} \times 5^8 \times 7^5 \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29 \times 31 \times 37$


Factorial of $52$

The number of ways there are to shuffle a $52$-card deck is given by:

$52! = 806 \ 58175 \ 17094 \ 38785 \ 71660 \ 63685 \ 64037 \ 66975 \ 28950 \ 54408 \ 83277 \ 82400 \ 00000 \ 00000$


Prime Factors of $52!$

The prime decomposition of $52!$ is given as:

$52! = 2^{49} \times 3^{23} \times 5^{12} \times 7^8 \times 11^4 \times 13^4 \times 17^3 \times 19^2 \times 23^2 \times 29 \times 31 \times 37 \times 41 \times 43 \times 47$


Factorial of $450$

\(\ds 450!\) \(=\) \(\ds 17333 \, 68733 \, 11263 \, 26593 \, 44713 \, 14610 \, 45793 \, 99677 \, 81126 \, 52090\)
\(\ds \) \(\) \(\ds 51015 \, 56920 \, 75095 \, 55333 \, 00168 \, 34367 \, 50604 \, 67508 \, 82904 \, 38710\)
\(\ds \) \(\) \(\ds 61458 \, 11284 \, 51842 \, 40978 \, 58618 \, 58380 \, 63016 \, 50208 \, 34729 \, 61813\)
\(\ds \) \(\) \(\ds 51667 \, 57017 \, 19187 \, 00422 \, 28096 \, 22372 \, 72230 \, 66352 \, 80840 \, 38062\)
\(\ds \) \(\) \(\ds 31236 \, 93426 \, 74135 \, 03661 \, 01015 \, 08838 \, 22049 \, 49709 \, 29739 \, 01163\)
\(\ds \) \(\) \(\ds 67937 \, 66165 \, 02373 \, 08538 \, 96403 \, 90159 \, 08361 \, 44149 \, 59443 \, 26842\)
\(\ds \) \(\) \(\ds 04513 \, 78471 \, 64023 \, 03182 \, 60409 \, 46839 \, 93315 \, 06130 \, 25639 \, 18385\)
\(\ds \) \(\) \(\ds 30334 \, 15106 \, 06761 \, 46242 \, 02058 \, 20006 \, 93635 \, 20959 \, 67417 \, 18319\)
\(\ds \) \(\) \(\ds 15387 \, 25617 \, 50952 \, 13805 \, 56781 \, 30919 \, 54298 \, 00229 \, 27380 \, 33425\)
\(\ds \) \(\) \(\ds 53558 \, 16459 \, 19962 \, 98912 \, 36859 \, 85477 \, 71179 \, 15846 \, 13513 \, 40068\)
\(\ds \) \(\) \(\ds 90564 \, 71276 \, 58164 \, 83637 \, 71263 \, 03774 \, 92336 \, 00780 \, 72307 \, 46200\)
\(\ds \) \(\) \(\ds 85543 \, 55068 \, 36144 \, 81266 \, 06281 \, 14576 \, 09604 \, 99187 \, 81342 \, 83979\)
\(\ds \) \(\) \(\ds 24840 \, 59250 \, 45378 \, 49487 \, 42506 \, 04884 \, 81036 \, 57144 \, 79570 \, 46788\)
\(\ds \) \(\) \(\ds 63574 \, 29367 \, 14615 \, 17621 \, 91484 \, 69743 \, 10297 \, 99497 \, 40714 \, 48510\)
\(\ds \) \(\) \(\ds 47161 \, 69664 \, 05239 \, 73926 \, 02848 \, 40869 \, 40074 \, 08998 \, 90112 \, 74929\)
\(\ds \) \(\) \(\ds 05171 \, 51447 \, 34313 \, 86633 \, 39249 \, 20406 \, 61522 \, 69230 \, 30438 \, 13960\)
\(\ds \) \(\) \(\ds 54196 \, 60932 \, 24243 \, 80922 \, 51372 \, 68851 \, 71790 \, 43032 \, 14058 \, 23844\)
\(\ds \) \(\) \(\ds 79361 \, 11678 \, 56823 \, 69730 \, 36238 \, 40462 \, 65078 \, 90688 \, 00000 \, 00000\)
\(\ds \) \(\) \(\ds 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000\)
\(\ds \) \(\) \(\ds 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 00000 \, 0\)


Factorial of $1\,000$

The factorial of $1\,000$ starts:

$402,387,260,077 \ldots$

and has $2568$ digits, of which the last $249$ are $0$.


Factorial of $1\,000\,000$

The factorial of $1 \, 000 \, 000$ has $5 \, 569 \, 709$ digits.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Factorial/Examples&oldid=738885"