The coefficient of x²⁰¹² in the expansion (1 – x)²⁰⁰⁸ (1 + x + x² )²⁰⁰⁷ is equal to

The correct answer is 0.

We need to find the coefficient of x²⁰¹² in the expansion of (1 - x)²⁰⁰⁸(1 + x + x²)²⁰⁰⁷.

The key insight is to use the algebraic identity:

(1-x)(1+x+x^2)=1-x^3

We can rewrite the given expression by splitting the exponent on (1 - x)²⁰⁰⁸:

(1-x){}^{2008}(1+x+x^2){}^{2007}=(1-x)·(1-x){}^{2007}·(1+x+x^2){}^{2007}

Grouping the last two factors using our identity:

=(1-x)·{\left[\right(1-x\left)\right(1+x+x^2\left)\right]}^{2007}=(1-x)·(1-x^3){}^{2007}

Now we need the coefficient of x²⁰¹² in (1 - x)(1 - x³)²⁰⁰⁷.

Expanding this product:

\left[\mathrm{coeff.\; of}{x}^{2012}\mathrm{in}\right(1-x^3\left){}^{2007}\right]-\left[\mathrm{coeff.\; of}{x}^{2011}\mathrm{in}\right(1-x^3\left){}^{2007}\right]

The general term in the expansion of (1 - x³)²⁰⁰⁷ is:

{(-1)}^k\left(\genfrac{}{}{0ex}{}{2007}{k}\right)x^{3k}

This means every non-zero term has a power of x that is a multiple of 3.

Check for x²⁰¹²:
We need 3k = 2012, which gives k = 2012/3 = 670.666... — not an integer.
⇒ Coefficient of x²⁰¹² in (1 - x³)²⁰⁰⁷ = 0

Check for x²⁰¹¹:
We need 3k = 2011, which gives k = 2011/3 = 670.333... — not an integer.
⇒ Coefficient of x²⁰¹¹ in (1 - x³)²⁰⁰⁷ = 0

Therefore, the coefficient of x²⁰¹² in the original expression is:

0-0=0