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Forum URL: http://www.cut-the-knot.org/cgi-bin/dcforum/forumctk.cgi
Forum Name: Middle school
Topic ID: 48
Message ID: 2
#2, RE: 4!!
Posted by Tom on Oct-10-01 at 08:00 PM
In response to message #0
>I need to know what 4!! equals

n!! = (2^(n/2)) * (n/2)! for even values of n,

or n!! = n!/(n-1)!! for odd values of n

This means that 4!! = 8, using the first equation

> and how you reached the answer

example: 12!! = 12*10*8*6*4*2, or

(2)(6)*(2)(5)*(2)(4)*(2)(3)*(2)(2)*(2)(1), or

(2^6)*(6*5*4*3*2*1) = (2^6)(6!) = (2^(12/2))*(12/2)!

= (2^(n/2))*(n/2)!

Try this for other even numbers too, they all work.

For odd values of n, for example 11,

11!! = 11*9*7*5*3 = (11*10*9*8*7*6*5*4*3*2)/(10*8*6*4*2)

but the denominator is just (11-1)!! which can be found from the first equation. So the complete form for odd numbers is

n!/(n-1)!! and n-1 will be even if n is odd.

This isn't a formal proof but intuitively it works.