Nice explanation! It's implied in your example that the one same (evidently indestructible) deck of cards is used throughout the 100 year lifespan.
But what if you always start with a deck where the cards are in the exact same order to begin with? Say, the order of a standard deck of Bicycle cards when it is first opened?
If you took every deck of bicycles, for however long they have been packaging their cards in the same order, that has ever been opened (I'll let you calculate what that number might be)
and then shuffle each of those decks only FOUR times (as he does in the video), i bet you might get more than a few repeated sequences.
Now, I won't claim that he started with a brand new unshuffled deck, since he flashes the four of spades on the bottom before he starts shuffling. But i would wager that it takes shuffling any new deck more than four times to achieve the full range of values of 52 factorial....
just sayin'... 
Edit: Couldn't resist googling this myself, and apparently people much better at math then myself have determined that 7 shuffles is the minimum number to "approach" true randomness.
And this is only true for "good" shuffling methods... One author says quote "By the way, the overhand shuffle is a really bad way to mix cards: it takes about 2500 overhand shuffles to randomize a deck of 52 cards!"
The reason for this has to do NOT with the total possible number of permutations, but with the number of "rising sequences in the permutation"...
Here's a link to the full math explanation for those of you who can handle it (Not me!)
http://www.dartmouth.edu/~chance/teaching_aids/Mann.pdf
Actually, there's no such implication at all. It need not be the same deck with every shuffle.
Well, when you say "shuffle a deck more than a thousand times a day" i kind of think it does
imply shuffling the same deck over and over, whether you meant it that way or not. I'm sure you'll be quick to correct me, but assumptions will be made by your audience if you don't stipulate, and it's always easier to "clarify" after the fact. In any case, that's not the point...
My point was that one cannot say that "no two shuffled decks in the history of cards have ever been in the same order". Which is actually a subtly different point from what they said in the video.
I still say that if you always start with the "new deck sequence" and you shuffle only four times, and you repeat
this process millions of times, it is probable that you will have some non-unique sequences.
It doesn't matter the condition in which the deck started. You could reorder the deck every single time to "factory-fresh" order, but as long as you completely randomize the deck, the odds of repeating a previous shuffle are still the same.
You say it doesn't matter what order the deck starts in as long "as you completely randomize the deck", which you later say requires at least seven shuffles...?
So in other words... it does matter what order the deck starts in, if you are only shuffling it four times...
All I am trying to say boils down to just this: "I bet most people who open a new deck of cards don't shuffle it well enough the first time to achieve a truly random permutation...i.e. seven times or more...".
And to my knowledge, Professor/Magician Persi Diaconis of Stamford University was the first to discover that a deck required at least seven thorough riffle shuffles in order to be randomized, preferably more.
Yes, Diaconis. The link I included refers to the analysis by Bayer and Diaconis on the first page, and is an extensive exposition of the math involved. It's kind of fun to try to follow along...
And while 7 is Diaconis' preferred number of shuffles, he does claim that as few as 4 may be enough for games like blackjack where the suits are inconsequential. Other mathematicians have come up with even higher numbers of shuffles than 7, based on different ways of measuring and quantifying "randomness".
Cheers!
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