[Siegmund]

[ QUOTE ]
I know the number of possible shuffles of a 52 card deck is 52!. How many of these shuffles are, for all intents and purposes, basically the same?

[/ QUOTE ]

Define what your intents and purposes are and what "basically the same" is.

In stuffy mathematicsese: your notion of "basically the same" defines an equivalence relation. That equivalence relation breaks up the set of 52! permutations into some number of equivalence classes.

In holdem, we don't care about the 4! permutations of suits, which of two hole cards is dealt first to each player, what order the three flop cards are in, or what order the stub cards are in. In a 10-handed game, it looks like there are 52!/(2^10 * 3! * 1 * 1 * 27! * 4!) ~ 50234623599558030217607550528000000 unique deals, but you could probably argue there are somewhat fewer than that e.g. by tracking down the situations where flushes are completely impossible.

In stud it's harder - we sometimes care about the suits (if we have to choose which of two equal cards brings it in) and sometimes don't - so the sets of equivalent deals are not equal in size.

In bridge it's easier - we always care about suits, but never care about what order each player receives his 13 cards.