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

Answer: (A)

0

Answer 2

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