Prove that $(2n)!$ is always divisible by $n!(n+1)!$.
Context: I am a beginner in number theory. To build up my foundation I am following the book Higher Algebra by S. Barnard and J.M. Child.
I know that the problem could be easily solved by doing simple algebraic manipulation and arguing that Catalan numbers are always integers, but I am wondering if there is another approach to attack this problem, particularly using Legendre's formula.