In trials the probability that the first are successes and the remaining trials are failures is .

Now the probability of successes and failures in some other (specified order) is the same as this. So to get the probaility that we need ( successes and failures) irrespective or order we need to multiply by the number of distinct permutations of objects of one type and of the other where objects of the same type are indistinguishable. Which is where the binomial coefficient comes from.

RonL