I haven’t found any neat approximations, despite a quick search on Google.
Best estimate I have seen is 8x10^67
Approximations are often turned into time lines assuming you shuffle&deal the pack once every second !
Just looked up that there are 1.33x10^50 atoms in the Earth. Age of universe is 13.8bn years. So if there was a deck of playing cards for every atom on Earth each being dealt every second since the Big Bang, it would still take another 5bn years to reach 52! deals. Even then there’s a good chance that any particular deal would not have occurred.
Paraphrasing Douglas Adams:
“Dealing cards is big, really big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way across space, but that’s just peanuts to dealing cards.”
52! is such an easy number to quote, and in the context of a deck of cards so easy to grasp the concept of shuffling and dealing - but so difficult to imagine just how big the number of permutations there are.
I carefully explained this visualisation of dealing 52 cards in all their different ways to Mrs D.
I had to take it slowly, especially the concept of how many atoms there are in the Earth - she could visualise the back garden (1/3 acre) - but was struggling a bit with the Earth.
She didn’t feel that atoms could shuffle and deal a deck of cards every second- they would need at least a minute ! and what had the Big Bang to do with it - nobody was around at that point in time
But eventually, I thought i’d managed to deliver a reasonable concept of how vastly, hugely, mind-bogglingly big it is until she concluded with “so not really worth trying, I suppose ?”
My ballpark estimate of 2x10^70 was disappointingly 250 times too high.
Stirling’s approximation (I cheated by looking it up) is (sqrt(2n x pi)) x (n/e)^n, giving 8.0529x10^67. I’m not sure that’s any easier than multiplying out all the numbers. Maybe it was designed for larger n than 52.
If anyone needs an exact answer, I got this from Xnumbers (a free 3rd party add-in for Excel):
That’s likely Excel’s default formatting. The calculations should be good for 15 or 16 figures if reformatted.
Xnumbers (powerful but clunky to use) can in theory correctly maintain anything up to about 30,000 significant figures, though it’s rather meaningless to try and see them all!