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If the coefficients of $r^{t h}$ and ${(r + 1)}^{t h}$ terms in the expansion of ${(1 + x)}^{24}$ are in the ratio 12 : 13, then r is the root of the quadratic equation

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$x^{2} - 5 x + 6 = 0$

$x^{2} - 11 x + 30 = 0$

$x^{2} - 14 x + 13 = 0$

$x^{2} - 14 x + 24 = 0$

answer is D.

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Detailed Solution

$\frac{24_{C_{r - 1}}}{24_{C_{r}}} = \frac{12}{13} \Rightarrow \frac{24_{C_{r}}}{24_{C_{r - 1}}} = \frac{13}{12} \Rightarrow \frac{24 - r + 1}{r} = \frac{13}{12} \Rightarrow r = 12 ∴ 12 {is a root ofx}^{2} -12x+24$ $x^{2} - 14 x + 24 = 0$

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