I have given $N$ I.I.d. Random variables $\{X_i\}_{i=1,\dots,N}$, distributed according to $\mathbb{P}[X=x]=f(x)$ which has support on the Real positive line and a parameter $C\in\mathbb{R}^+$. I draw a realisation of these $N$ variables and I order them from the smallest to the largest: $\xi_1<\xi_2<\dots<\xi_N$ ($\xi$ is a realisation of the random variable $X$). What is the probability that $C$ is larger than $\xi_k$ but smaller than $\xi_{k+1}$, that is $\mathbb{P}[\xi_1<\xi_2<\dots<\xi_k<C<\xi_{k+1}<\dots<\xi_N]?$
Edit
I share my answer. $\mathbb{P}[\xi_1<\xi_2<\dots<\xi_k<C<\xi_{k+1}<\dots<\xi_N]$ is the probability that exactly $k$ realisations are smaller than $C$ and $k+1$ are larger, this can be computed using a binomial distribution: $$ \mathbb{P}[\xi_1<\xi_2<\dots<\xi_k<C<\xi_{k+1}<\dots<\xi_N]=\binom{N}{k}\Big(\mathbb{P}[X>C]\Big)^{N-k}\Big(\mathbb{P}[X<C]\Big)^{k}\,. $$
Can anyone confirm this solution?