Question

The coefficient of $x^{203}$ in the expression $(x-1)(x^2-2)(x^3-3)\cdots (x^{20}-20)$ is

• A. $11$
• B. $12$
• C. $13$
• D. $15$

Answer 1

The correct option is C $13$

Let $E=(x-1)(x^2-2)(x^3-3)\cdots (x^{20}-20)$
$\Rightarrow E=x(1-\frac{1}{x})\cdot x^2(1-\frac{2}{x^2})\cdot x^3(1-\frac{3}{x^3})\cdots x^{20}(1-\frac{20}{x^{20}})$
$\Rightarrow E=x^{210}(1-\frac{1}{x})(1-\frac{2}{x^2})\cdots (1-\frac{20}{x^{20}})$

Now, coefficient of $x^{203}$ in $E$ is coefficient of $x^{-7}$ in $(1-\frac{1}{x})(1-\frac{2}{x^2})\cdots (1-\frac{20}{x^{20}})$

Coefficient of $x^{–7}$
$=-7+1\cdot 6+2\cdot 5+3\cdot 4-1\cdot 2\cdot 4$
$=-7+6+10+12-8$
$=13$