A question asks:
Random samples $X_1, X_2, ..., X_n$ have the probability density function
$f(x)=1, 0<x<1$ and $f(x) = 0$ otherwise.
They are sorted such that $X_{(1)} <X _{(2)}<...<X_{(n)}$. Find the probability density function of $X_{(k)}$.
The answer begins with $$f_{x_{(k)}}(x) = \frac{n!}{(k-1)!(n-k)!} f(x)[F(x)]^{k-1}[1-F(x)]^{n-k}$$
It looks like a binomial distribution to me, but seems incredibly odd that in a binomial distribution, the Combination Function is $\frac{n!}{(n-k)!k!}$. Why then, would there be a $(k-1)!$ ? Also, why do I have to multiply in an additional $f(x)?$ into the entire function?
Any help is much appreciated! Thank you!