Question

Find the coefficient of $x^{11}$ in ${(2x^2+\frac{3}{x^3})}^{13}$

Answer 1

${({2x}^2+\frac{3}{x^3})}^{13}$

$\Rightarrow {(r+1)}^{th}$ term $={{}^{13}C}_r{\left(2x^2\right)}^{13-r}{\left(\frac{3}{x^3}\right)}^r$

$\Rightarrow {{}^{13}C}_r{2}^{13-r}\times 3^r\times x^{26-5r}$

As we want $x^{11}$ so, $x^{26-5r}=x^{11}$

$\Rightarrow x^{26-5r}=x^{11}$

i.e, $26-5r=11$

$\Rightarrow 5r=15$

$\Rightarrow r=3$

So, ${{}^{13}C}_3\times 2^{13-3}\times 3^3$

$\Rightarrow \frac{13!}{10!3!}\times 2^{10}\times 27$

$\Rightarrow \frac{11\times 12\times 13}{6}\times 2^{10}\times 27$

$\Rightarrow 7722\times 2^{10}.$

Hence, the answer is $7722\times 2^{10}.$