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 December 8th, 2011, 06:07 AM #11 Member   Joined: Nov 2011 From: Vietnam Posts: 41 Thanks: 0 Re: Law of Factorial 1. The Formula of Factorial Law: n! = Sum_{i=0...n}(-1)^i*C(n,i)*(n+1-i)^n is giving some insights about factorial. It is right the property of factorial. The Formula of Factorial Law shows us n! is a sum of n+1 terms (i=0...n) while the primitive Formula of Factorial Definition shows us n! is a product of n factors (n! = 1×2×...×n). 12! is a sum of 13 terms (i=0...12) 12! = 23 298 085 122 481 -106 993 205 379 072 +207 136 272 863 586 -220 000 000 000 000 +139 802 620 558 095 -54 425 825 574 912 +12 789 349 373 724 -1 724 011 610 112 +120 849 609 375 -3 690 987 520 +35 075 106 -49 152 +1 = 383 147 212 602 368 -383 146 733 600 768 = 479 001 600 2. The “n!/i!(n-1)!” when n=12 and i=7 is “12!/7!(12-7)!=12!/7!5!” and is symbolized only, but in fact equals to 12×11×...×1/(7×6×...×1)(5×4×...×1)=12×11×...×8/5×4×...×1=792 or 12×11×...×1/(7×6×...×1)(5×4×...×1)=12×11×...×6/7×6×...×1=792 So there is no problem about calculating n! though inside the formula there are some factorials too. 3. The Formula of Factorial Law...“which makes it look a lot less like a factorial formula” but why is that! Because it is an other factorial formula, and right is the answer for the question: “Can you find the factorial of a given natural number n without using the definition formula of factorial?” (only use it whenever testing a result). 4. By the Law of Factorial: n! is a sum of n+1 terms (i=0...n) 12! is a sum of 13 terms (i=0...12) 11! is a sum of 12 terms (i=0...11) ... 3!=1×4^3 -3×3^3 +3×2^3 -1×1^3=6. 3! is a sum of 4 terms. 2!=1×3^2 -2×2^2 +1×1^2=2. 2! is a sum of 3 terms. 1!=1×2^1 -1×1^1=1. 1! is a sum of 2 terms. 0!=1×1^0=1. 0! is a sum of 1 term. 0!=1×1^0=1 is a proof of 0! = 1 being a Property of Factorial, Not need a Convention 0! = 1 nor a Definition 0! = 1.
 December 8th, 2011, 04:58 PM #12 Senior Member   Joined: Nov 2011 Posts: 595 Thanks: 16 Re: Law of Factorial Yes, but we don't need this to know that 0!=1. Actually $\Gamma(n)=(n-1)!$ and $\Gamma(1)=1$. Furthermore in your prood you are using the fact that $C^0_0=1$ which actually comes from the definition 0!=1. So it is the snake eating is own tail here! Also I agree with CRGreathouse . It should not actually be called law of factorial. Actually you can divide from both side by n!, then it gives you 1=some of stuffs, which to me looks more like a Taylor expansion of some function or some combinatory property, so to my point of view this is not directly related to factorial. However this is a nice formula, but I am not sure that it is not already known, I would have to spend more time to understand what it is exactly or look at your proof. cheers. Thanks from ducnhuandoan
 February 29th, 2016, 07:51 PM #13 Member   Joined: Nov 2011 From: Vietnam Posts: 41 Thanks: 0 Reply to Law of Factorial and Law of Exponentiation CRGreathouse Wrote: "I thought so at first -- neat, might give some insight into the factorial, but computationally worthless. But now I'm questioning "insight into the factorial". Not the insight part, not yet anyway, but the factorial part. When you think about it, every term on the right is of the form n! * (stuff) so perhaps this should be written 1 = sum ... instead... which makes it look a lot less like a factorial formula." All was written, by donoring a bit or Fair play to, you and everyone can see: https://www.smashwords.com/books/sea...ery=nhuan+doan Many Thanks
June 14th, 2016, 12:20 AM   #14
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the formula of factorial law and a "pregnant woman"

Quote:
 Originally Posted by Dougy Yes, but we don't need this to know that 0!=1. Actually $\Gamma(n)=(n-1)!$ and $\Gamma(1)=1$. Furthermore in your prood you are using the fact that $C^0_0=1$ which actually comes from the definition 0!=1. So it is the snake eating is own tail here! Also I agree with CRGreathouse . It should not actually be called law of factorial. Actually you can divide from both side by n!, then it gives you 1=some of stuffs, which to me looks more like a Taylor expansion of some function or some combinatory property, so to my point of view this is not directly related to factorial. However this is a nice formula, but I am not sure that it is not already known, I would have to spend more time to understand what it is exactly or look at your proof. cheers.
$n!=\sum_{i=0}^{n}(-1)^{i}*{\frac{n!}{(i)!(n-i)!}}*{(n+1-i)^{n}}$

The formula n! is the same as "a snake eating its tail!" â€“Dougy. But I see this formula n! the same as a "pregnant woman" is more correct. So, there is nothing more absurd than when someone asked why the fetus was lying in a pregnant woman's belly!

Last edited by ducnhuandoan; June 14th, 2016 at 12:31 AM.

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# natural factorial of 12 what is the natural factorial of 12

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