Question

Coefficient of x²⁰¹² in (1-x)²⁰⁰⁸(1+x+x²)²⁰⁰⁷ is equal to ___

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is A

Consider the expansion:
\((1 - x)^{2008} \quad \text{and} \quad (1 + x + x^2)^{2007}.\)

The expansion of \((1 - x)^{2008}\) yields terms of the form \((-1)^k \binom{2008}{k} x^k\) for \(k \geq 0\). Thus, it contains only terms with non-positive powers of \(x\) (i.e., \(x^0, x^1, x^2, \dots\)).

The expansion of \((1 + x + x^2)^{2007}\) contains terms of the form \(x^m\), where \(m\) is a non-negative integer ranging from 0 to 4014 (since the highest power in the expansion occurs when all factors contribute \(x^2\)).

To find the coefficient of \(x^{2012}\) in the product:  
\((1 - x)^{2008} \cdot (1 + x + x^2)^{2007},\)
we note that there is no term in \((1 - x)^{2008}\) with a negative power of \(x\) to combine with terms in \((1 + x + x^2)^{2007}\) such that the resulting power of \(x\) is 2012. Therefore, the coefficient of \(x^{2012}\) in the expansion is:  0

The correct option is (A) : 0