If $^{n^{2} - n} C_{2} =^{n^{2} - n} C_{10}$ , then n=

12

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4

only

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- 3

only

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44$ or

- 3

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Solution

The correct option is B $4$ only
We have
$^{n^{2} - n} C_{2} =^{n^{2} - n} C_{10}$
We know that
$^{n} C_{x} =^{n} C_{y} \Rightarrow x = y o r x + y = n$
$\Rightarrow 2 + 10 = n^{2} - n$
$\Rightarrow n^{2} - n = 12$
$\Rightarrow n^{2} - n - 12 = 0$
$\Rightarrow n^{2} - 4 n + 3 n - 12$
$\Rightarrow n (n - 4) + 3 (n - 4) = 0$
$\Rightarrow (n + 3) (n - 4) = 0$
$\Rightarrow n = 4, - 3$
So, $n = 4$ [ $n$ is non negative integer]

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