Consider a sample $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$. Let $T_n=F_n^{-1}(1/2)$ be the sample median where, $F^{-1}(x)=\inf\{t:F(t)\ge x\}$ and $F_n(y)=\frac{1}{n}\sum_{i=1}^n\mathbb{I}(X_i\le y)$. Assume that $N=2m-1$. Then, $T_n=X_{(m)}$.
Now, let $X^*_1,...X_n^*$ be an iid sample from $F_n$. Then, $$P_*(X_{(m)}^*=X_{(k)}|X_1,...X_n)=\sum_{j=0}^{m-1}{n\choose j}\frac{(k-1)^j(n-k+1)^{n-j}-k^j(n-k)^{n-j}}{n^n}.$$
[Ref. The Jackknife & Bootstrap (Jun Shao)]
I'm having trouble understanding how they arrived at this expression. Any hints on how to proceed?