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(C) Let $N = n+5.$The coefficients of three consecutive terms in the expansion of $(1+x)^N$ are given by $^N C_{r-1}, ^N C_r,$ and $^N C_{r+1}.$Given

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$462$

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(C) Let $N = n+5.$The coefficients of three consecutive terms in the expansion of $(1+x)^N$ are given by $^N C_{r-1}, ^N C_r,$ and $^N C_{r+1}.$Given the ratio $^N C_{r-1} : ^N C_r : ^N C_{r+1} = 5 : 10 : 14.$From $\frac{^N C_r}{^N C_{r-1}} = \frac{10}{5} = 2,$we have $\frac{N-r+1}{r} = 2$ $\Rightarrow N-r+1 = 2r$ $\Rightarrow N+1 = 3r. \quad (1)$From $\frac{^N C_{r+1}}{^N C_r} = \frac{14}{10} = \frac{7}{5},$we have $\frac{N-r}{r+1} = \frac{7}{5}$ $\Rightarrow 5N-5r = 7r+7$ $\Rightarrow 5N-12r = 7. \quad (2)$Substituting $r = \frac{N+1}{3}$ into $(2)$:$5N - 12(\frac{N+1}{3}) = 7$$5N - 4(N+1) = 7$$5N - 4N - 4 = 7 \Rightarrow N = 11.$Then $r = \frac{11+1}{3} = 4.$The expansion is $(1+x)^{11}.$ The largest coefficient is the middle term,which is $^{11} C_6 = \frac{11 	imes 10 	imes 9 	imes 8 	imes 7}{5 	imes 4 	imes 3 	imes 2 	imes 1} = 462.$

If for some positive integer $n,$ the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14,$ then the largest coefficient in this expansion is

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