query: Using permutation or otherwise, prove that $\frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.
positive_passages.docid: 20691
positive_passages.text: prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
negative_passages.docid: 86413
negative_passages.text: Show that $\frac{(m+n-1)!}{m!n!}$ is an integer where m and n are positive integers and gcd(m,n)=1
negative_passages.docid: 10629
negative_passages.text: How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction
negative_passages.docid: 34374
negative_passages.text: Simplifying factorials: $\frac{(n-1)!}{(n-2)!}$
negative_passages.docid: 87843
negative_passages.text: Combinatorial proof that $\frac{({10!})!}{{10!}^{9!}}$ is an integer
negative_passages.docid: 10604
negative_passages.text: Prove $3^n > n^2$ by induction
negative_passages.docid: 97684
negative_passages.text: Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$
negative_passages.docid: 26336
negative_passages.text: Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
negative_passages.docid: 89265
negative_passages.text: Proving that $x - \frac{x^3}{3!} 0$
negative_passages.docid: 141837
negative_passages.text: Proof of $\sup\{ (n!)^{(\frac{1}{n})}| n ∈ N\} = +∞?$
negative_passages.docid: 4750
negative_passages.text: Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.
negative_passages.docid: 7894
negative_passages.text: Does it follow that $(n!)^n$ divide $(n^2)!$
negative_passages.docid: 33810
negative_passages.text: Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer
negative_passages.docid: 75123
negative_passages.text: If $p$ be a prime and r be any integer, $0 < r < p$ then $\frac{(p-1)!}{r!(p-r)!}$ is an integer.
negative_passages.docid: 116901
negative_passages.text: Monotonicity of $\frac{n}{\sqrt[n]{(n!)}}$
negative_passages.docid: 121544
negative_passages.text: Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
negative_passages.docid: 143357
negative_passages.text: Comparing $2^n$ to $n!$