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प्रश्न
The coefficient of the term independent of x in the expansion of \[\left( ax + \frac{b}{x} \right)^{14}\] is
पर्याय
\[14! a^7 b^7\]
\[\frac{14!}{7!} a^7 b^7\]
\[\frac{14!}{\left( 7! \right)^2} a^7 b^7\]
\[\frac{14!}{\left( 7! \right)^3} a^7 b^7\]
उत्तर
\[\frac{14!}{\left( 7! \right)^2} a^7 b^7\]
\[\text { Suppose (r + 1)th term in the given expansion is independent of x } . \]
\[\text{ Then, we have} \]
\[ T_{r + 1} = ^{14}{}{C}_r (ax )^{14 - r} \left( \frac{b}{x} \right)^r \]
\[ = ^{14}{}{C}_r a^{14 - r} b^r x^{14 - 2r} \]
\[\text{ For this term to be independent of x, we must have } \]
\[14 - 2r = 0\]
\[ \Rightarrow r = 7\]
\[ \therefore \text{ Required term } = ^{14}{}{C}_7 a^{14 - 7} b^7 = \frac{14!}{(7! )^2} a^7 b^7\]
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