Question

The coefficient of x²⁵⁶ in the expansion of (1 – x)¹⁰¹(x² + x + 1)¹⁰⁰ is ______.

Options

• –¹⁰⁰C₁₆

• ¹⁰⁰C₁₆

• ¹⁰⁰C₁₅

• –¹⁰⁰C₁₅

Answer 1

The coefficient of x²⁵⁶ in the expansion of (1 – x)¹⁰¹(x² + x + 1)¹⁰⁰ is `underlinebb(""^100C_15)`.

Explanation:

(1 – x)¹⁰¹(x² + x + 1)¹⁰⁰ = (1 – x)¹⁰⁰(x² + x + 1)¹⁰⁰(1 – x)

= [(1 – x)(1 + x + x²)]¹⁰⁰(1 – x)

= (1 – x³)¹⁰⁰(1 – x)

= (1 – x)(¹⁰⁰C₀ – ¹⁰⁰C₁x³ + ¹⁰⁰C₂x⁶ – ..... + ¹⁰⁰C₈₄x²⁵² – ¹⁰⁰C₈₅x²⁵⁵ + ¹⁰⁰C₈₆x²⁵⁸ + ....)

∴ Coefficient of x²⁵⁶ is ¹⁰⁰C₈₅ = `""^100"C"_(100–85)` = ¹⁰⁰C₁₅