The coefficient of \(\frac{1}{\text{x}}\) in the expansion of…

The coefficient of \(\frac{1}{\text{x}}\) in the expansion of \((1+\text{x})^{\text{n}}+\Big(1+\frac{1}{\text{x}}\Big)^{\text{n}}\) is:

  1. None of these.

  2. \(\frac{(\text{2n})!}{\big[(\text{2n}-1)!(\text{2n}+1)!\big]}\)

  3. \(\frac{(\text{2n})!}{\big[(\text{n}-1)!(\text{n}+1)!\big]}\)

  4. \(\frac{\text{n}!}{\big[(\text{n}-1)!(\text{n}+1)!\big]}\)

Solution

Solution:

Coefficient of \(\frac{1}{\text{x}}\) in the given expansion = Coefficient of 1 in \((1+\text{x})^{\text{n}}\) × Coefficient of \(\frac{1}{\text{x}}\)

\(={^\text{n}}\text{C}_{\text{0}}\times{^\text{n}}\text{C}_{\text{1}}+{^\text{n}}\text{C}_{\text{1}}\times{^\text{n}}\text{C}_{\text{2}}\)

\(=\text{n}+\text{n}\times\frac{\text{n}!}{2(\text{n}-2)!}\)

\(=\text{n}+\text{n}\frac{\text{n}(\text{n}-1)}{2}\)

Asked in: RDSHARMA

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