Factorial/Examples

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

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): Tables: $5$ The Factorials of the Numbers $1$ to $20$
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): Tables: $5$ The Factorials of the Numbers $1$ to $20$