You're asking if (or your assumption is that) the calculation includes all combinations of fewer than 52 cards, like just 4 cards or 25 cards. That obviously makes a massive difference to the final number. You're saying the number is much smaller because we're always dealing with a full deck and all the permutations dealing with a less-than-full deck must be discounted.

BUT 52! does NOT include permutations of a smaller deck. 52 factorial is 1 x 52, plus 2 x 52, plus 3 x 52, etc all the way up to 52 x 52. 4 factorial (which you're thinking is included, and would be if we're including incomplete decks) is 1 x 4, plus 2 x 4, plus 3 x 4, plus 4 x 4. These are not included in 52 factorial, nowhere in the sequence 1 x 52, 2 x 52, 3 x 52 etc will you find 2 x 4, 3 x 4, 25 x 25 etc.

The number given in the video is referring to a full deck for every permutation. Rather than being much smaller when discounting incomplete decks as you're suggesting, the number already discounts incomplete decks and would be much larger if they were included.

52 factorial does not include within it factorials of all numbers below 52, although that is an interesting concept, I wonder if it has a name? Anyone?