Given a random variable $X_1$ drawn from a distribution with cdf $F$, and random variables $X_2, \cdots,X_n$ drawn from another distribution with cdf $G$, what is the formula for the probability that $X_1$ is the $k$-th out of the $n$ variables, when they are sorted in order. If it helps, assume that $F$ and $G$ are both defined on [0,1], with positive density everywhere.
By computing the probability that $k-1$ of the $G$-distributed $X$'s are less than or equal to $X_1$, conditional on the value of $X_1 = t$, and integrating over all $t$ we obtain
This of course assumes independence of all the $X_i$. Given no assumptions about any relationship about $F$ and $G$, this expression should not simplify in general.