Question

The coefficient of $x^{10}$ in $(1+x^2-x^3{)}^8$ is

• A. $476$
• B. $496$
• C. $506$
• D. $528$

Answer 1

Answer: (A)

$476$

In $(1+x^2-x^3{)}^8$
There are only two possible ways of $x^{10}$
Those are $\left(x^2{)}^5\right(1^3\left)\right(-x^3{)}^0$ and $\left(x^2{)}^2\right(-x^3{)}^2)1^4$
Hence the corresponding coefficients will be
$\frac{8!}{5!(3!)}$ and $\frac{8!}{2!(2!)4!}$
Thus the coefficient of $x^{10}$ in the above expression is
$\frac{8!}{5!(3!)}+\frac{8!}{2!(2!)4!}$
$=56+420$
$=476$