Question

The number of distinct terms in the expansion of ${(\sqrt{x}+\frac{1}{\sqrt{x}}+x^{\frac{3}{2}}+\frac{1}{x^{\frac{3}{2}}})}^{40}$ is/are (with respect to different power of $x$)

• A. $120$
• B. $121$
• C. $60$
• D. $31$

Answer 1

The correct option is B $121$

${(\sqrt{x}+\frac{1}{\sqrt{x}}+x^{\frac{3}{2}}+\frac{1}{x^{\frac{3}{2}}})}^{40}$
$={\left(\frac{x^2+x+x^3+1}{x^{\frac{3}{2}}}\right)}^{40}$
$=\left(\frac{a_0+a_{1}x+a_2{x}^2+a_3{x}^3+.....+a_{120}{x}^{120}}{x^{60}}\right)$