Im struggling to solve this question (The context is Catalan numbers): With only "up" and "right" moves from $(0,0)$ to $(n,n)$, We'll call the "up" moves above the $y=x$, "Bad Moves".
Let $0 \leq k \leq n-1$,
Prove that the number of routes with (k) Bad Moves = the number of routes with (k+1) Bad Moves.