If C6 n+2P2 n-2=11, then n satisfies the equation :

C6 n+2P2 n-2=11,n+2!n-4!6!=11n-2!n-4!n+2n+1nn-1n-2n-3=11×6!×n-2n-3n+2n+1nn-1=11×10×9×8n=9

If C6 n+2P2 n-2=11, then n satisfies the equation :

C6 n+2P2 n-2=11,n+2!n-4!6!=11n-2!n-4!n+2n+1nn-1n-2n-3=11×6!×n-2n-3n+2n+1nn-1=11×10×9×8n=9

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If $\frac{{}^{n + 2}C_{6}}{{}^{n - 2}P_{2}} = 11,$ then n satisfies the equation :

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$n^{2} + 3 n - 108 = 0$

$n^{2} + n - 110 = 0$

$n^{2} + 2 n - 80 = 0$

$n^{2} + 5 n - 84 = 0$

answer is C.

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Detailed Solution

$\frac{{}^{n + 2}C_{6}}{{}^{n - 2}P_{2}} = 11,$

$\frac{(n + 2)!}{(n - 4)! 6!} = 11 \frac{(n - 2)!}{(n - 4)!}$

$(n + 2) (n + 1) n (n - 1) (n - 2) (n - 3) = 11 \times 6! \times (n - 2) (n - 3)$

$(n + 2) (n + 1) n (n - 1) = 11 \times 10 \times 9 \times 8$

$n = 9$

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If $\frac{{}^{n + 2}C_{6}}{{}^{n - 2}P_{2}} = 11,$ then n satisfies the equation :