Question

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

• A. 300
• B. 200
• C. 100
• D. 201

Answer 1

Answer: (D)

201

Answer 2

$\left\lbrack 1 + \left( x^{3} + \frac{1}{x^{3}} \right) \right\rbrack^{100}$1 + ${ } ^ { 100 } \mathrm { C } _ { 1 }$ $\left( x^{3} + \frac{1}{x^{3}} \right)$

+${ } ^ { 100 } \mathrm { C } _ { 2 }$ $\left( x^{3} + \frac{1}{x^{3}} \right)^{2}$+…….+ ${ } ^ { 100 } \mathrm { C } _ { 100 }$ $\left( x^{3} + \frac{1}{x^{3}} \right)^{100}$

= (1 + r) + a₁ x³ + a₂x⁶ +…… + a₁₀₀(x³)¹⁰⁰ + b₁$\frac{1}{x^{3}}$+ …….+b₁₀₀$\left( \frac{1}{x^{3}} \right)^{100}$

all other terms obtained by combination of x³ and $\frac{1}{x^{3}}$ well get converted into a term involving x³ or $\frac{1}{x^{3}}$ and hence it will be present among above terms

So number of terms = 1 + 100 + 100 = 201