Question

The number of terms in ${(x^3+1+\frac{1}{x^3})}^{100}$ is

• A. 300
• B. 200
• C. 100
• D. 201

Answer 1

The correct option is D

201

${[1+(x^3+\frac{1}{x^3})]}^{100}=1{+}^{100}{C}_1(x^3+\frac{1}{x^3}){+}^{100}{C}_2{(x^3+\frac{1}{x^3})}^2+\cdots {+}^{100}{C}_{100}{(x^3+\frac{1}{x^3})}^{100}$
$=(1+r)+\alpha 1x^3+\alpha 2x^6+\cdots +\alpha 100(x^3{)}^{100}+\beta 1\frac{1}{x^3}+\cdots +\beta 100{\left(\frac{1}{x^3}\right)}^{100}$; r = constant coming from combination of terms.
All other terms obtained by combination of $x^3$ and $\frac{1}{x^3}$ well get converted into a term involving $x^3$ or $\frac{1}{x^3}$ and hence it will be present among above terms
So number of terms = 1 + 100 + 100 = 201