If $^{2 n} C_{3}$ : $^{n} C_{2}$ :: $44 : 1$ , the value of $n$ is

17

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6

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11

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none of these

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Solution

The correct option is A $17$
$^{2 n} C_{3} :^{n} C_{2} :: 44 : 1$

$\Rightarrow \frac{\frac{2 n!}{(2 n - 3)! 3!}}{\frac{n!}{(n - 2)! 2!}} = \frac{44}{1}$
$\Rightarrow \frac{2 n!}{n!} \times \frac{(n - 2)! 2!}{(2 n - 3)! 3!} = \frac{44}{1}$
$\frac{2 n!}{n!} \frac{(n - 2)! 2!}{(2 n - 3)! 3 \times 2!} = \frac{44}{1}$

$\Rightarrow \frac{2 n (2 n - 1) (2 n - 2)}{3 n (n - 1)} = \frac{44}{1} \Rightarrow \frac{2 n (2 n - 1) .2 (n - 1)}{3 n (n - 1)} = \frac{44}{1}$
$\Rightarrow \frac{2.2 n (2 n - 1)}{3 n} = \frac{44}{1}$
$\Rightarrow \frac{4 n (2 n - 1)}{3 n} = \frac{44}{1} \Rightarrow 2 n^{2} - n = 33 n$
$\Rightarrow 2 n^{2} = 34 n$
$\Rightarrow 2 n^{2} - 34 n = 0$
$= 2 n (n - 17) = 0$

$\Rightarrow n = 0, 17$

$n = 17$

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If 2nC3:nC2::44:1, the value of n is

If $^{2 n} C_{3}$ : $^{n} C_{2}$ :: $44 : 1$ , the value of $n$ is