Let n=2015 The least positive integer k for which kn2n2−12n2−22n3−32…n2−(n−1)2=r! for some positive integer r is

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

We can rewrite the given expression as

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

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

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

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

Related content