I was reading about factorials recently, and I happened to come across the curious, almost pseudo-Pythagorean-seeming fact that $$6!7!=10!$$ I was greatly intrigued by this, but couldn't think of any justification other than "it just happens to be that way".
Yes, I have seen this question and answer. I understand that it's not exactly known when or where these crop up in general, and in fact only a few are generally known. I'm just asking about whether, for this particular case, there is a reason why this fact "should" be the case beyond just the relatively prosaic fact that $6!=8\times 9\times 10$, the missing factors from $7!$.
So my question is: Is it a fact that seems coincidental but actually has a very good rationale, like that $e^{\pi\sqrt{163}}\approx 262537412640768743.999999999999\approx 262537412640768744$, which has to do with Heegner numbers and a bunch of other things I don't quite understand? Or is it a pure coincidence like the fact that $e^{\pi}-\pi\approx 19.999099979\approx 20$?