(Thoughts) Set of tuples? Same elements? One-to-One Correspondence? (Solution) Let $B(n, r, k)$ the set of all tuples that satisfies the given conditions. For convenience, Let $B(st, s, t), B(st-s, s, t), B(st-t, s, t)$ $\alpha, \beta, \gamma$ respectively. We will show that $|\alpha|=|\beta|=|\gamma|$. Set an arbitrary element of $\alpha$ $(x_{1}, \cdots , x_{s})$. If we subtract 1 from all $x$s, it becomes an element of $\beta$. We can easily check that $x_{s}>0$ by the 2nd and 3rd conditions. Reversely, all elements of $\beta$ correspond to an element of $\alpha$. This proves the first equal sign. Similar correspondence can be found on $\gamma$ by $(x_{2}, \cdots , x_{s}, x_{1}-t)$. It's not hard to check that all 3 conditions are met. Reverse process is also possible, so $|\alpha|=|\gamma|$ also holds. This concludes the proof. (Miscellaneous things) A similar one was on 2018 KMO... I think? Something as $a+2b+2c+d=n$ and random conditions.