Question

Coefficient of x²⁰³ in the expression

(x – 1) (x² – 2) (x³ – 3)..... (x²⁰ – 20) must be-

• A. 11
• B. 12
• C. 13
• D. 15

Answer 1

Answer: (C)

13

Answer 2

Expression = x . x². x³..... x²⁰$\left( 1 - \frac{2}{x^{2}} \right)$

$\left( 1 - \frac{3}{x^{3}} \right)$......$\left( 1 - \frac{20}{x^{20}} \right)$ = x²¹⁰ . E

Where E = $\left( 1 - \frac{1}{x} \right)$ $\left( 1 - \frac { 2 } { \mathrm { x } ^ { 2 } } \right)$ $\left( 1 - \frac{3}{x^{3}} \right)$......$\left( 1 - \frac{20}{x^{20}} \right)$

Now coefficient of x²⁰³ in original expression = coefficient of x^–7 in E

But E = 1 – $\left( \frac{1}{x} + \frac{2}{x^{2}} + \frac{3}{x^{3}} + ..... \right)$

+$\left( \frac{1}{x}.\frac{6}{x^{6}} + \frac{2}{x^{2}}.\frac{5}{x^{5}} + \frac{3}{x^{3}}.\frac{4}{x^{4}}..... \right)$– $\left( \frac{1}{x}.\frac{2}{x^{2}}.\frac{4}{x^{4}}..... \right)$

= Coefficient of x^–7

= –7 + 6 + 10 + 12 – 8 = 13.