How to Evaluate 4!!!!

Date: 10/16/2003 at 17:38:14
From: liz
Subject: 4!!!!

Is there a resource in any books that could give me the exact value
for 4!!!!?  Our Calculus 2 teacher has challenged us to find the
answer, but the number is too large to reasonably calculate by hand:

   4! = 24
   24! = 

I really appreciate any help that you could provide!

Date: 10/16/2003 at 19:07:43
From: Doctor Shawn
Subject: Re: 4!!!!

Liz,

4!!!! is too big to even reasonably calculate with good computer
software.  You might want to check out this archived answer:

  Googol!
    http://mathforum.org/library/drmath/view/57903.html 

The thing is, your number is going to be much larger even than any of 
the numbers under discussion.  24! is about 6.20448402 x 10^23; taking
the factorial of that explicitly is more or less beyond any feasible
computer or calculator; and taking the factorial of THAT explicitly
would require more time than can exist.

I think the purpose of this assignment is to show you how fast the 
factorial function explodes.  You can use the techniques on that page 
to attempt an approximation, but it won't be very accurate.

Does that help?
 
- Doctor Shawn, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/17/2003 at 12:41:49
From: Doctor Douglas
Subject: Re: 4!!!!

Hi Liz,

Thanks for writing to the Math Forum.

I'd like to supplement Dr. Shawn's answer.  There are different ways
to interpret multiple factorial signs.  One way, of course, is

  3!!! =  ((3!)!)! = ((6)!)! = 720! 

Another way is to do the following:

  n! = n(n-1)(n-2)...              the usual factorial

  n!! = n(n-2)(n-4)...             e.g. 5!! = 5*3*1 = 15
                                   "double factorial"
  n!!! = n(n-3)(n-6)...

  n!!!! = n(n-4)(n-8)...

etc., where the products run only over positive integers.  This is
called "multifactorial" notation.  You can read more about it here:

  Eric Weisstein's World of Mathematics -- Multifactorial
    http://mathworld.wolfram.com/Multifactorial.html 

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/20/2003 at 03:54:48
From: liz
Subject: Thank you (4!!!!)

Thanks for the info!