ETA: If you want to avoid your phony problem with permutations and combinations, go back to my example of shuffling a regular deck of cards and dealing out 5 cards. Whatever 5 cards you get is a unique combination, and the odds are something like 1:2.5 million. All of them--even the "hands" that lack any apparent pattern.
You're confusing permutations and combinations. All permutations are equally likely, but that is not true of all combinations. Which is why just about everyone on earth outside this Forum would be astonished by the combination of 1000 heads and no tails while yawning at the overwhelming majority of combinations of 500 heads and 500 tails.
No I'm not. You're lumping together thousands of possible outcomes and comparing the probability of getting one of those to getting exactly one outcome.
As I mentioned, I could do the same with the all heads outcome by making the category "all outcomes that begin with H", and the odds are 1:2.
But you knew that already.
Can you say why the outcome of 500 heads in a row followed by 500 tails in a row is not as "meaningful" as 1000 heads?
Or the pattern "HTHTHTHT. . . "
Or the pattern "HHHHTTTTHHHHTTTT. . ."
All these three are possible outcomes that include 500 heads and 500 tails. Surely that's not the same as some patternless result that happens to total 500 of each, at least as regards to the synchronicity game.
Do you mean to tell me you would find 1000 consecutive heads (on an honest coin with fair tosses) is a message from the universe, but getting exactly 4 heads in a row then 4 tails in a row over and over for all 1000 tosses is not?
My point is that with this synchronicity business you're not saying it's only the 1000 heads outcome that is meaningful. That determination is arbitrary and made after the fact. As such, it is simply the Texas Sharpshooter Fallacy.
In fact, I've adequately explained how this works. Our brains evolved to spot patterns. When we see patterns in random data it is a Type I error, and it's called apophenia.