Re factorials versus binomial coefficients --- `left as they are' assumes that the expression has been arrived at in the first place using the kind of argument you have been employing; but that is not necessarily the case.
You mentioned factorials as a possible source of problems for the formulae I was offering. Those formulae (though no doubt wrong, and for all I know unfixable), did not contain any factorials.
At the same time, as I was trying to say earlier, I suspect that reasoning about these problems intuitively, though seductive, is ultimately both limited and error-prone. For instance, your current approach is vitiated by the lack of any formal calculus to identify reliably those path shapes which are relevant.
That would be why I've repeatedly said I'm only guessing at the answer, may well be wrong, etc. You've now protested twice at my alleged appeals to intuition. The first time, what I gave was a rigorous proof with no appeals to intuition in it. The second time, I wasn't purporting to offer anything other than handwaving. I agree that there is such a thing as unsound appeal to intuition, and I'm sure I've been guilty of it many a time -- but in the present instance I must demur. -- g