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File: Calculating52FactorialByHand

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!! Calculating 52 Factorial By Hand - 2016/01/04

Some time ago I gave a talk in which I showed that something unexpected
happened with a deck of playing cards.  I had some volunteers try it,
and while they did so I talked about just how many orderings there are
for 52 cards.  I computed (an approximation to) $52!$ (52 factorial) by
hand. It's not so hard - you just calculate $54!$ and then divide by 3000.

I know that sounds like a joke, but it's not.  Here's why.

Firstly, I'm going to be really rough and ready here, and we can come
back later to refine our calculations.  So I'll start with Stirling's
approximation to the factorial:

|>> $n!~{\approx}~(n/e)^n\sqrt{2{\pi}n}$ <<|

We can use that to compute $52!,$ but suddenly we notice that $e$ is
roughly $2.7.$  That means that if we compute $54!$ we get:

|>> $54!~\approx~(54/e)^{54}\sqrt{2{\pi}54}$ <<|

And $54/e$ is roughly $20,$ so this simplifies enormously.  Taking just
the first part:

|>> $(54/e)^{54}~{\approx}~20^{54}$ <<|

We can break that down further:

|>> $20^{54}=2^{54}10^{54}$ <<|

And

|>> $2^{54}=2^4(2^{10})^5~{\approx}~16\cdot(10^3)^5=16\cdot10^{15}.$ <<|

Trace that back, re-insert it all, and we have:

|>> $(54/e)^{54}~{\approx}~(16\cdot10^{15})(10^{54})=16\cdot10^{69}$ <<|

Now we return to $\sqrt{2{\pi}54}.$ Using $pi\approx3$ this simplifies
to $\sqrt{324}$ which is 18.  Pulling it all together, we get:

|>> $54!~{\approx}~16\cdot10^{69}\cdot18=288\cdot10^{69}$ <<|

But now we want to divide by $(53\times54),$  which is close enough to $2880,$
so we get:

|>> $52!~{\approx}~(2880\cdot10^{68})/2880=10^{68}$ <<|

And there we are - we've computed (an approximation to) $52!$ by computing
$54!$ and dividing by $3000.$

Obvious, really.

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