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Odd factorials *April 12, 2007*

*Posted by eric22222 in General, Math.*

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WARNING! MATH-HEAVY POST!

My mind was wandering off while riding in the car on Wednesday. I started thinking about an uncommonly used mathematical formula: factorials. An exclamation mark indicates a factorial. The value of x! would be the product of every value from x to 1. For example:

5! = 5*4*3*2*1 = 120

Using multiple factorials, you can go down to values other than one:

7! / 3! = 7*6*5*4*~~3*2*1~~ / ~~3*2*1~~ = 7*6*5*4 = 840

Anyway, I got to thinking how it’d be possible to make a factorial that somehow skips even or odd numbers. Eventually I thought up a formula for it. For even numbers:

(x/2)! * 2^(x/2)

where x is the even number you want. Plugging in 8, for example, we get:

(8/2)! * 2^(8/2) = 4! * 2^4 = 4*3*2*1 * 2*2*2*2 = 8*6*4*2 = 384

For odd numbers, we need the normal factorial divided by the even factorial. With x as the highest odd number:

x! / (((x-1)/2)! * 2^((x-1)/2))

Trying 7, we get:

7! / (((7-1)/2)! * 2^((7-1)/2)) = 7! / ((6/2)! * 2^(6/2)) = 7! / (3! * 2^3)

= 7! / (3*2*1 * 2*2*2) = 7*~~6~~*5*~~4~~*3*~~2~~*1 / ~~6*4*2~~ = 7*5*3*1 = 105

Not sure if these formulas will ever come in use for anybody, but if does, hooray! I’m useful!

This is a great solution!

Hooray! Extraordinarily useful, thanks! You are brilliant and I really admire you.