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

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

1. A

2014

2. B

2013

3. C

1

4. D

2

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### Solution:

We can rewrite the given expression as

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

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

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

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

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