Question

The coefficient of $x^{1012}$ in the expansion of $(1+x^n+x^{253}{)}^{10}$, (where $n\le 22$ is any positive integer), is :

• A. $1$
• B. ${}^{10}{C}_4$
• C. $4n$
• D. ${}^{253}{C}_4$

Answer 1

The correct option is B ${}^{10}{C}_4$

$T_{r+1}{=}^{10}{C}_r{(1+x^n)}^{10-r}{{(x}^{253})}^r$
$\Rightarrow 253r=1012$
$\Rightarrow r=4$
Coefficient is ${}^{10}{C}_4$ .