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(C) Let the three successive terms be $T_{r+1}, T_{r+2},$ and $T_{r+3}.$ Their coefficients are $^{n}C_{r}, ^{n}C_{r+1},$ and $^{n}C_{r+2}.$Given the

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$7$

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(C) Let the three successive terms be $T_{r+1}, T_{r+2},$ and $T_{r+3}.$ Their coefficients are $^{n}C_{r}, ^{n}C_{r+1},$ and $^{n}C_{r+2}.$Given the ratio $^{n}C_{r} : ^{n}C_{r+1} : ^{n}C_{r+2} = 1 : 7 : 42.$From $\frac{^{n}C_{r+1}}{^{n}C_{r}} = \frac{7}{1},$ we have $\frac{n-r}{r+1} = 7 \implies n-r = 7r+7 \implies n = 8r+7.$From $\frac{^{n}C_{r+2}}{^{n}C_{r+1}} = \frac{42}{7} = 6,$ we have $\frac{n-(r+1)}{r+2} = 6 \implies n-r-1 = 6r+12 \implies n = 7r+13.$Equating the two expressions for $n$: $8r+7 = 7r+13 \implies r = 6.$The first of these terms is $T_{r+1} = T_{6+1} = T_{7},$ which is the $7^{th}$ term.

If the coefficients of the three successive terms in the binomial expansion of $(1 + x)^n$ are in the ratio $1 : 7 : 42,$ then the first of these terms in the expansion is (in $^{th}$)

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