You mean, like why does the binomial distribution calculation come out like that?

If you have n trials with k successes at p probability, then the event will happen with p^k probability, and not happen with (1-p)^(n-k) probability. (We usually call 1-p q to simplify that.) We multiply those, since both occur.

And since these can happen in any order, we want to account for that, by counting all the orders. Thats nCk ("n choose k"), the binomial coefficient as a multiplier. That's the ways to choose k things from a set of n, which makes a certain amount of sense.