Why binomial distribution doesn't count permutations? Why in Binomial distribution the formula starts with $n\choose k$ and not with something like $k!\over n!$? Isn't the order important? Or, it is important but due to the independence of the event?   

$$\frac{n!}{n_1!n_2!},\;\; n_1+ n_2=n,$$
counts the permutations of the sequence
$$\underbrace{p_1...p_1}_{n_1}\underbrace{p_2...p_2}_{n_2}.$$
Coincidentally, $$\frac{n!}{n_1!n_2!}=\frac{n!}{k!(n-k)!}={n \choose k}, \;\; n_1=k=\text{number of successes}.$$
Therefore, ${n \choose k}$, disguised as counting combinations, actually counts permutations.
