If ^2n C_ 3 : ^n C_ 2 =44:3, find n. - Vidyadip

Question

If ${^\text{2n}}\text{C}_{\text{3}}:{^\text{n}}\text{C}_{\text{2}}=44:3,$ find n.

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Answer

We have,
$\Rightarrow \frac{\frac{2\text{n}!}{(3\text{n})!(2\text{n-3}!)}}{\frac{\text{n}!}{\text{2}!(\text{n-2}!)}}$
$\Rightarrow \frac{2\text{n}!2!(\text{n}-2)!}{3!(2\text{n}-3)!\text{n}!}=\frac{44}{3}$
$\Rightarrow \frac{2\text{n}!}{3\text{n}~(\text{n-1})!(2\text{n}-3)!}=\frac{44}{3}$
$\Rightarrow 2\text{n}(2\text{n}-1)(2\text{n}-2)=44\text{n}(\text{n}-1)$
$\Rightarrow (2\text{n}-1)(\text{n}-1)=11(\text{n}-1)$
$\text{n}=6$

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