Let $X_1,\dots,X_n$ be iid random continuous variables with cdf $F$ and $X_{(1)}\leq \dots\leq X_{(n)}$ their order statistcs.
I know that $\mathbb{P}(X_{(r)}\leq u)=F_{X_{(r)}}(x) = \sum_{j=r}^n {n\choose j}F(x)^j(1-F(x))^{n-j}$.
I would like to express $\mathbb{P}(X_{(r+1)}\geq u\geq X_{(r)})$ in term of the $F(x)^j$, is there a simple expression ?
The event is equivalent to there are $r$ of them less than $u$, and the remaining $n-r$ of them greater than $u$. So the probability is simply
$$ \binom {n} {r} F(u)^r [1 - F(u)]^{n - r}$$
The binomial coefficient count the number of ways to partition the group.