The coefficient of the term independent of x in the expansion of…

The coefficient of the term independent of x in the expansion of \(\Big(\text{ax}+\frac{\text{b}}{\text{x}}\Big)^{14}\) is:

  1. \(\frac{14!}{7!}\ \text{a}^{7}\ \text{b}^{7}\)

  2. \(\frac{14!}{(7!)^{2}}\ \text{a}^{7}\ \text{b}^{7}\)

  3. \(\frac{14!}{(7!)^{3}}\ \text{a}^{7}\ \text{b}^{7}\)

  4. \(14!\ \text{a}^{7}\ \text{b}^{7}\)

Solution

Solution:

Suppose (r + 1)th term in the given expansion is independent of x.

Then, we have

\(\text{T}_{\text{r}+1}={^\text{14}}\text{C}_{\text{r}}(\text{ax})^{14-\text{x}}\ \Big(\frac{\text{b}}{\text{a}}\Big)^{\text{r}}\)

\(={^\text{14}}\text{C}_{\text{r}}\ \text{a}^{14-\text{r}}\ \text{b}^{\text{r}}\ \text{x}^{14-2\text{r}}\)

For this term to be independent of x, we must have

\(=14-2\text{r}=0\)

\(\Rightarrow \text{r}=7\)

Required term \(= {^\text{14}}\text{C}_{\text{7}}\ \text{a}^{14-7}\ \text{b}^{7}=\frac{14!}{(7)!}\ \text{a}^{7}\ \text{b}^{7}\)

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