Question

The coefficient of $x^{203}$ in $(1-x)(2-x^2)(3-x^3)\cdots (20-x^{20})$ is
(correct answer + 1, wrong answer - 0.25)

• A. $11$
• B. $21$
• C. $13$
• D. $15$

Answer 1

The correct option is C $13$

We have,
$(1-x)(2-x^2)(3-x^3)\cdots (20-x^{20})$
$=(x-1)(x^2-2)(x^3-3)\cdots (x^{20}-20)$
Highest possible coefficient of $x$ in above expansion is $x^{210}.$

Consider integer partitions with distinct part of $210-203=7.$

The ways of writing $7$ as a sum of distinct positive integers are $7,6+1,5+2,4+3,4+2+1$

So, in order to obtain $x^{203}$ coefficient, $x^7$ coefficient from the product of terms has to be excluded.
$\therefore$ Required coefficient is given by $-7+1\cdot 6+2\cdot 5+3\cdot 4-1\cdot 2\cdot 4=13$