Question

Coefficient of $x^{2001}$ in the expansion of $(1+x+x^2+x^3+x^4{)}^{1001}(1-x{)}^{1002}$ is

• A. $0$
• B. $4{}^{1001}{C}_{501}$
• C. $-2009$
• D. $2{\times }^{1001}{C}_{501}$

Answer 1

The correct option is A $0$

In binomial expansion of negative index,

$(1-x{)}^{-1}=(1+x+x^2+......x^n{)}^{1001}$ is

$\Rightarrow (1+x+x^2+x^3+....x^n{)}^{1001}(1-x{)}^{1002}$

$=(1-x{)}^{-1001}(1-x{)}^{1002}$

$=(1-x)$

$\Rightarrow$ coefficient of $x^{2001}$ is $\mathrm{0.....}$