These two formulas give n! This was discovered by Euler. Reference this link for further reading. http://eulerarchive.maa.org/hedi/HEDI-2007-09.pdf ... If I am not mistaken, Manjul Bhargava generalized the factorial function to include numbers such as (pi)! More on math.stackexchange.com

n! is simply defined as n(n-1)(n-2)...(3)(2)(1). This is done to shorten expressions, since this type of thing is used a lot in combinatorics. Simply stated, for n objects, there are n! ways to arrange it in a sequence. Like, for example if you have three balls of colours red, orange and blue, the possible sequences are: Red, Orange, Blue Red, Blue, Orange Orange, Blue, Red Orange, Red, Blue Blue, Red, Orange Blue, Orange, Red And so you can see, for 3 objects, there were 3! = 3 * 2 * 1 = 6 ways to arrange it. Now, 0! = 1 because the math works out that way, but an intuitive (and kinda philosophical) way to think about it is that if you have 0 objects, how many ways can you arrange them? Well, that's 1, because it's the arrangement of having 'nothing'. I'll go into the math a little bit here. We'll denote the number of permutations of n different objects taken r at a time by nP(r) to make life easier. So, for example, if we have the 4 balls, A, B, C, D, but we instead take two at a time (or 4P(2)), then there are the following ways to arrange it: A, B A, C A, D B, A B, C B, D C, A C, B C, D D, A D, B D, C So, that's 12 ways to arrange it. You might note that's 4 * 3, so we can work out nP(r) to be: (n)(n-1)(n-2)...(n-r+1). By definition, n! = n * (n-1)!, so this can also be stated as: (n!)/(n-r)! Now, we have 4P(4). Well, obviously that's going to be 4 * 3 *2 * 1 or 4!, but by our formula, we have 4!/0!. So, we need 4!/0! = 4!, or 0! = 1. So, in order to ensure nP(n) = n!, we define 0! = 1 More on reddit.com

The answer is complicated. Essentially, there is a way to extend the factorial function to all of the complex plane except the negative integers called the Γ function . To understand why this is the "right" extension one needs to learn some complex analysis. In essence, this is the only "nice" function that agrees with the factorial on positive integer (well, kinda, due to technical reasons its more comfortable to define Γ such that it satisfies a historical mistake Γ is defined such that Γ(n)=(n-1)!). However, the extension of factorial specifically for half positive integers (e.g. to 1/2) was worked out before the full blown Γ function was defined (so I imagine that late 19th century mathematicians were very pleased to find that defining Γ this way is consistent with the definition I'm about to describe): it essentially followed from working out the volumes and areas of n-dimensional balls and spheres. People noted that for even values of n, the volume of an n-dimensional ball is pi^(n/2)/(n/2)! and the surface area of an n-dimensional ball (that is, an (n-1)-dimensional sphere) is 2pi^(n/2)/(n/2-1)!. It then made sense to define (n/2)! for general n as V(n)/pi^(n/2) where V(n) is the folume of an n-dimensional ball. Once doing so, it became apparent that the properties of factorial are preserved for this extension, for example (n/2)!(n/2+1) = (n/2+1)!. More on reddit.com

  log (n!) = log [n * (n - 1) * (n - 2) * ... * 1], n things are being multiplied.
              >= log [n * (n - 1) * ... * (n / 2)], n/2 things are being multiplied here, so I'm just dropping the second half.
              >= log [(n / 2)^(n / 2)], because each one is greater than or equal to n / 2.
              = n/2 log (n / 2).

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