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Combination

If n + 2C8 ...

Question

If $(n + 2) C_{8} : (n - 2) P_{4} = 57 : 16$ , find $n$ .

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Solution

Given that,

$\frac{(n + 2) C_{8}}{(n - 2) P_{4}} = \frac{57}{16}$

$⟹ \frac{\frac{(n + 2)!}{(n - 6)! 8!}}{\frac{(n - 2)!}{(n - 6)!}} = \frac{57}{16}$

$⟹ \frac{(n + 2) (n + 1) (n) (n - 1)}{8!} = \frac{57}{16}$

$⟹ (n + 2) (n + 1) (n) (n - 1) = 143640 = 21.20.19.18$

$⟹ n = 19$

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