Question

The term independent of $x$ in the expansion of ${\left( {1 + x} \right)^n}{\left( {1 + \dfrac{1}{x}} \right)^n}$ is

1. $C_0^2 + C_1^2 + C_2^2 + ... + C_n^2$
2. ${}^{2n}{C_n}$
3. $\dfrac{{1 \cdot 3 \cdot 5 \cdot ... \cdot \left( {2n - 1} \right)}}{{n!}}{2^n}$
4. All of the above

Answer 1

Hint: The statement of the binomial theorem tells us how to expand expressions of the form, that is, {\left( {a + b} \right)^n} = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}} n \\\ k \end{array}} \right){a^{n - k}}{b^k}} , where $a$ and $b$ are the terms, $n$ is the total number of possibilities and $k$ is the required number of possibilities. Use this binomial theorem, and then use the given conditions from the given question to find the required value.

Complete step-by-step answer:

Given expression is ${\left( {1 + x} \right)^n}{\left( {1 + \dfrac{1}{x}} \right)^n}$.

Simplifying the above expression to find the form for the above stated binomial theorem, we get

$$\Rightarrow {\left( {1 + x} \right)^n}{\left( {\dfrac{{1 + x}}{x}} \right)^n} \\\ \Rightarrow {\left( {1 + x} \right)^n}\dfrac{{{{\left( {1 + x} \right)}^n}}}{{{x^n}}} \\\ \Rightarrow \dfrac{{{{\left( {x + 1} \right)}^{2n}}}}{{{x^n}}} \\\ \Rightarrow {x^{ - n}}{\left( {1 + x} \right)^{2n}} \\\$$

We know that the binomial theorem tells us how to expand expressions of the form, that is, {\left( {a + b} \right)^n} = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}} n \\\ k \end{array}} \right){a^{n - k}}{b^k}} , where $a$ and $b$ are the terms, $n$ is the total number of possibilities and $k$is the required number of possibilities.

Now, we will now use the above binomial theorem to know that the required term independent of $x$ is equals to the coefficient of ${x^0}$ in ${x^{ - n}}{\left( {1 + x} \right)^{2n}}$.

{x^{ - n}}{\left( {1 + x} \right)^{2n}} = \sum\limits_n^{2n} {\left( {\begin{array}{*{20}{c}} {2n} \\\ n \end{array}} \right){1^{2n - n}}{x^n}} {x^{ - n}} \\\ = {}^{2n}{C_n}{x^0} \\\

Since we know that the coefficient of ${x^0}$ in ${x^{ - n}}{\left( {1 + x} \right)^{2n}}$ is equals to the coefficient of ${x^n}$ in ${\left( {1 + x} \right)^{2n}}$ using the above stated binomial theorem.

We will now find the coefficient of ${x^n}$ in ${\left( {1 + x} \right)^{2n}}$ to find the term, which is independent of $x$ from the given expression using the binomial theorem.

{\left( {1 + x} \right)^{2n}} = \sum\limits_n^{2n} {\left( {\begin{array}{*{20}{c}} {2n} \\\ n \end{array}} \right){1^{2n - n}}{x^n}} \\\ = {}^{2n}{C_n}{x^n} \\\

Thus, we get that the term independent of $x$ in the expansion of ${\left( {1 + x} \right)^n}{\left( {1 + \dfrac{1}{x}} \right)^n}$ is ${}^{2n}{C_n}$.
Hence, option B will be correct.

Note: In solving these types of questions, students must note that when they are asked to find the term independent of some variable, you should always put the power of the variable equals 0. Also, we should take special care while writing each term and cross-check if we have written it correctly or not. Students should use the binomial theorem to reduce some steps for finding the solution.