Question

Find the coefficient of $x^{12}$ in $(1+x^2{)}^4(1+x^3{)}^7(1+x^4{)}^{12}$

Answer 1

$\text{First term}$

The highest power of the expression is $8$

So, the coefficient is of $x^{12}$ is $0$

$\text{Second term}$

General term is ${}^n{C}_r{x}^r$

Here $x$ is replaced by $x^3$

So taking $r$ to be $4$

${}^7{C}_4=35$

$\text{Third term}$

General term is ${}^n{C}_r{x}^r$

Here $x$ is replaced by $x^4$

So taking $r$ to be $3$

${}^{12}{C}_3=220$

so the coefficient of $x^{12}$ is $220+35=255$