$\text{ Let }n=2015$ The least positive integer $k$ for which $k\left(n^2\right)\left(n^2-1^2\right)\left(n^2-2^2\right)\left(n^3-3^2\right)\dots \left(n^2-(n-1{)}^2\right)=r!$  for some positive integer$r$ is

We can rewrite the given expression as 

$k\left(n^2\right)(n-1)(n+1)(n-2)(n+2)(n-3)(n+3)\dots$…….$(n+n-1)(n-n+1)=r!$

$\begin{array}{l}\Rightarrow kn\left(1\right)\left(2\right)\dots (n-1)n(n+1)(n+2)\dots (2n-1)=r!\\ \Rightarrow kn(2n-1)!=r!\end{array}$

$\therefore$ To convert L.H.S. to a factorial, we shall require

$k = 2$ which will convert it into $\left(2n\right)!$