Question

Coeffecient of $x^{18}$ in ${(1+x+{2x}^2+{3x}^3+\cdots +{18x}^{18})}^2$ is equal to

• A. $995$
• B. $1005$
• C. $1235$
• D. none of these

Answer 1

The correct option is B $1005$

In the expansion, $x^{18}$ will be formed when two terms are multiplied whose powers add up to 18.
Thus all terms of the form $x^k{x}^{18-k};0\le k\le 18$
Clearly, coefficients of such terms form the series $S_n=1\times 18+1\times 17+2\times 16+...+17\times 1+18\times 1$
$S_n=2(1\times 18)+{\sum }_{t=1}^{17}t(18-t)$
$=36+{\sum }_{t=1}^{17}(18t-t^2)$
$=36+18\times \frac{17\times 18}{2}-\frac{17\times 18\times 35}{6}$
$=36+2754-1785=1005$