Question

If the sum of the binomial coefficients in the expansion of ${(x+\frac{1}{x})}^n$ is $64$, then the term independent of '$x$' is equal to

• A. $10$
• B. $20$
• C. $40$
• D. $60$

Answer 1

The correct option is B $20$

Sum of all binomial coefficients$={}^n{C}_o+{}^n{C}_1+{}^n{C}_2+........+{}^n{C}_n=64$

But, ${}^n{C}_o+{}^n{C}_1+{}^n{C}_2+.......+{}^n{C}_n=2^n$

$\therefore 2^n=64$

$\Rightarrow n=6$

${(x+\frac{1}{x})}^6$

$p=x$, $q=\frac{1}{x}$

$T_{r+1}={}^n{C}_r{p}^r{q}^{n-r}$

$={}^6{C}_2{x}^2{\left(\frac{1}{x}\right)}^{6-r}$

${=}^6{C}_2{x}^{r+r-6}$

for term independent of x, power of $x=0$

$\therefore 2r-6=0$

$\therefore r=3$

$\therefore$ Term independent x

${=}^6{C}_3$

$=\frac{6\times 5\times 4}{3\times 2\times 1}=20$.