Consider a function $f\left(x\right)=\left(1-\frac{1}{x}\right)$ Then term independent of x in the expansion of $\left(f\right(x){)}^n\cdot {\left(f\left(-\frac{1}{x}\right)\right)}^n\text{ is }$

he given question is same as to find the term independent of x in the expansion of.

$(1+x{)}^n\cdot {\left(1-\frac{1}{x}\right)}^n$

The given expression can be written as

$=(-1{)}^n\cdot \frac{{\left(1-x^2\right)}^n}{x^n}$

The term independent of x in the above expansion is 6. (c) The expression is (1 + x) 101 (1-x + x 2 ) 100 same as the coefficient of$x^n\text{ in }(-1{)}^n\cdot {\left(1-x^2\right)}^n$ which in turn is equal to the coefficient of $x^n$ in

$\mathbf{(}\mathbf{-}\mathbf{1}{\mathbf{)}}^{\mathbf{n}}\{{ }^n{C}_0{+}^n{C}_1(-x{)}^2{+}^n{C}_2{\left(-x^2\right)}^2+ \dots {+}^n{C}_n{\left(-x^2\right)}^n\}$

and the expansion contains only even powers of x. So, if n is odd

$\Rightarrow$Coefficient is zero.