What’s the probability of a deck of 52 cards being in a particular order after a shuffle?
I presume it’s quite a big number?
It’s 52!, pronounced ‘52 factorial’, which is 52 * 51 * 50 * … * 1. This is because, to begin with, you have 52 possible cards to be in the top spot, then 51 possible for the next-to-top, then 50 for third-from-top, down to one possibility to be the very bottom of the deck.
This works out to 80658175170943878571660636856403766975289505440883277824000000000000 distinct arrangements of cards for a 52-card deck. Now, who cuts?
Many thanks - I had an inkling it might be that but the numbers were getting so huge I figured I’d slipped somewhere!
Would I express that as 8x10[sup]66[/sup] ?
edit:
I see from google that 52 ! = 8.06581752 × 10[sup]67[/sup]
If it makes things more apparent, I’ve seen things like 10x10[sup]12[/sup], where you make sure the exponent is always a multiple of three; this is usually called ‘engineering notation’.
Anyway, it doesn’t especially matter here, as the number is really too big to be readily comprehended and we aren’t comparing it to anything else.
The number can be compared and comprehended:
52! ~= 10^68
This is about equal to the number of atoms in our galaxy.
I prefer that over scientific notation, simply because the exponents that are a multiple of three have names. 26x10[sup]9[/sup] Hz is 26 GHz; 850x10[sup]-9[/sup] m is 850 nm.
I don’t think there’s a name for 10[sup]66[/sup] so for a number that big it doesn’t really matter to me.
So if I am dealt a Royal Flush twice in a row after a shuffle, it’s probably not due to random chance? 
It depends if someone was stacking the deck or not 
The odds of that are more than 1/52! – it’s only 10 cards, rather than 52 cards, and the order of the first 5 and the 2nd 5 don’t matter – but it’s still highly unlikely unless someone has stacked the deck.
I think the odds of that are
20/52 * 4/51 * 3/50 * 2/49 * 1/48
which comes out to one in 649,740. That’s WAY less than 52! !
ETA: that’s the probability of getting dealt a royal flush once. Twice on a given two hands would be one in 422 billion.
Also, there are 4 different suits in which you could deal a Royal Flush.
There are [sub]52[/sub]C[sub]5[/sub] = 2,598,960 different 5-card hands you could be dealt. Of these, exactly 4 are royal flushes. So the probability of your random, five-card hand being a royal flush is 4/2598960 – which is, as CurtC says, one in 649,740.