If C 3 : C 3 = 10 : 1 , n 2 n then the ratio ( n 2 + 3 n ) : ( n 2 - 3 n + 4 ) is [2023]

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Solution

Q.
If ${}^{2 n}{C_{3} : {}^{n}{C_{3} = 10 : 1,}}$ then the ratio $(n^{2} + 3 n) : (n^{2} - 3 n + 4)$ is [2023]

1 27 : 11

2 65 : 37

3 2 : 1

4 35 : 16

Ans.
(3)

We have, ${}^{2 n}{C_{3}} :^{n} C_{3} = 10 : 1$

$\Rightarrow \frac{(\frac{2 n!}{(2 n - 3)! 3!})}{\frac{n!}{3! (n - 3)!}} = \frac{10}{1}$

$\Rightarrow \frac{2 n (2 n - 1) (2 n - 2)}{6} \times \frac{6}{n (n - 1) (n - 2)} = \frac{10}{1}$

$\Rightarrow \frac{4 (2 n - 1)}{(n - 2)} = 10 \Rightarrow 8 n - 4 = 10 n - 20$

$\Rightarrow 16 = 2 n \Rightarrow n = 8$ $∴$ $(n^{2} + 3 n) : (n^{2} - 3 n + 4) = (64 + 24) : (64 - 24 + 4)$

$= 88 : 44 = 2 : 1$

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