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Question
In the expansion of (1 + x)n the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.
Solution
\[\text{ Suppose the three consecutive terms are } T_{r - 1} , T_r \text{ and } T_{r + 1} . \]
\[\text{ Coefficients of these terms are } ^{n}{}{C}_{r - 2} , ^{n}{}{C}_{r - 1} \text{ and } ^{n}{}{C}_r , respectively . \]
\[\text{ These coefficients are equal to 220, 495 and 792 } . \]
\[ \therefore \frac{^{n}{}{C}_{r - 2}}{^{n}{}{C}_{r - 1}} = \frac{220}{495}\]
\[ \Rightarrow \frac{r - 1}{n - r + 2} = \frac{4}{9}\]
\[ \Rightarrow 9r - 9 = 4n - 4r + 8\]
\[ \Rightarrow 4n + 17 = 13r . . . \left( 1 \right)\]
\[\text{ Also } , \]
\[\frac{^ {n}{}{C}_r}{^ {n}{}{C}_{r - 1}} = \frac{792}{495}\]
\[ \Rightarrow \frac{n - r + 1}{r} = \frac{8}{5}\]
\[ \Rightarrow 5n - 5r + 5 = 8r\]
\[ \Rightarrow 5n + 5 = 13r\]
\[ \Rightarrow 5n + 5 = 4n + 17 \left[ \text{ From Eqn} \left( 1 \right) \right]\]
\[ \Rightarrow n = 12\]
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