### Calculations - Factorials By Repeated Multiplication

To check that 12! solves 3. in the Applications part of Size we could find the value by repeated multiplication.

12! = 12(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)1 = 132(10)9! = 1,320(72)7! = 95,040(7)6! = 665,280(6)(5)4! =

(665,280)(30)(24) = (665,280)(24)(30) = 30(15,966,720) = 479,001,600

The number we are seeking to bound by the closest lower factorial value is almost ten times as large. It is:

4,700,063,497

However, the next factorial, 13!, is much larger than that.

Computer programs can yield the two factorial bounds for the very large number in 4. (the source of that question's difficulty is its size),

63130707451134435989380140059866138830623361447484274774099906755

Paul Leyland submitted the information:

factorial(51) = 1551118753287382280224243016469303211063259720016986112000000000000

and factorial(50) = 30414093201713378043612608166064768844377641568960512000000000000

Several other individuals also found the values 50! and 51!.

Tom Womack confirmed the 12!, 13! and 50!, 51! bounds. He found Gamma(13.8917534369) and "roughly Gamma(51.186119476)" as answers for problem 7. in Size. (Note that the Definition of the function causes Gamma of (n+1) to be n!, at integer values n.) Womack stated "I found the gamma values using the Maple 7 computer-algebra package, with the command

a:=63130707451134435989380140059866138830623361447484274774099906755; fsolve(GAMMA(b)=a, b=50..52);

The departmental server at Nottingham took about half a second to compute the answer to 100 decimal places (the first few are 51.18611947585356463740969346988174712620584773795606397346755995)."
 19 Feb 2002 Version http://www.cs.ucla.edu/~klinger/size.html ©2002 Allen Klinger