Question

The number of distinct terms in the expansion of ${(x^3+1+\frac{1}{x^3})}^n;x\in R^{+}$ and $n\in N$ is

• A. $2n$
• B. $3n$
• C. $2n+1$
• D. $3n+1$

Answer 1

Answer: (C)

$2n+1$

${(x^3+1+\frac{1}{x^3})}^n={(1+(x^3+\frac{1}{x^3}))}^n$

${=}^n{C}_0{+}^n{C}_1(x^3+\frac{1}{x^3})+..{+}^n{C}_n{(x^3+\frac{1}{x^3})}^n$

All the terms are distinct with powers

${\left(x^3\right)}^0,\left(x^3\right),{\left(x^3\right)}^2,...{\left(x^3\right)}^n,{\left(x^3\right)}^{-n},...{\left(x^3\right)}^{-1}$

Hence $(2n+1)$ terms.