First slide

Permutations

Question

$Let n = 2015$ 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

Moderate

A

2014
B

2013
C

1
D

2

Solution

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 $(2 n)!$

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