If ( 3 ^ 6 4 ^ 4 )k is the term, independent of x, in the binomial...

Given that ${\left(\dfrac{x}{4} - \dfrac{12}{{x}^{2}}\right)}^{12}$

${r}^{th}$ term of an expression is ${T}_{r + 1} = Cr n {x}^{n - r}{y}^{r}$ ${T}_{r + 1} = Cr 12 {\left(\dfrac{x}{4}\right)}^{12 - r}{\left(- \dfrac{12}{{x}^{2}}\right)}^{r} = Cr 12 {\left(\dfrac{1}{4}\right)}^{12 - r} ( - 12{)}^{r} \cdot {x}^{12 - r - 2r}$
$⇒ 12 - 3r = 0$
$⇒ r = 4$

So term independent from $x$ is ${T}_{5} = C4 12 \cdot \dfrac{1}{{4}^{8}} \cdot {3}^{4} \cdot {4}^{4} = C4 12 \cdot \dfrac{{3}^{4}}{{4}^{4}}$

We know that $Cr n = \dfrac{n !}{\left(n - r\right) ! r !}$

$= 55 \times \dfrac{{3}^{6}}{{4}^{4}}$

They given that it is $= k\dfrac{{3}^{6}}{{4}^{4}}$

So required value of $k = 55 .$