Question

The coefficient of $x^{203}$ in the expression $(x-1)(x^2-2)(x^3-3).........(x^{20}-20)$ must be ?

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

Answer 1

The correct option is C $13$

First find the maximum power of $x$ that is $\sum _{n=1}^{n=20}n=210$

Now, list all the possible combination to get $203,$ they are

$203=210-\left(7\right)$

$=210-(1+6)$

$=210-(2+5)$

$=210-(3+4)$

$=210-(1+2+4)$

Now find coefficient of $x^{203}$

In $7$

$(x^7-7)(x^{203}+a{x}^{202}+.....)=-7$

In $1,6$

$(x^1-1)(x^6-6)(x^{203}+a{x}^{202}+....)=6$

In $2,5$

$(x^2-2)(x^5-5)(x^{203}+a{x}^{202}+....)=10$

In $3,4$

$(x^3-3)(x^4-4)(x^{203}+a{x}^{202}+...)=12$

In $1,2,4$

$(x^1-1)(x^2-2)(x^4-4)(x^{203}+a{x}^{202}+...)=-8$

$\Rightarrow$ So the coefficient of $x^{203}$ $=-7+6+10+12-8=13$