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If \left(\frac{3^{6}}{4^{4}}\right) \mathrm{k} ,is the term, independent of x in the binomial expansion of \left(\frac{x}{4}-\frac{12}{x^{2}}\right)^{12}, then \mathrm{k} is equal to_________.
 

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General Term T_{r+1}= ^{12}C_{r}\left ( \frac{x}{4} \right )^{12-r}\left ( \frac{-12}{x^{2}} \right )^{r}
Term independent of x
12-r-2r= 0\Rightarrow r= 4
T_{5}= \frac{12!}{4!8!}\times \left ( \frac{1}{4} \right )^{8}\times \left ( -12 \right )^{4}
      = \frac{12\times 11\times 10\times 9}{4\times 3\times 2\times 1}\times \frac{12\times 12\times 12\times 12}{4\times 4\times 4\times4\times4\times4\times4\times4}
      = 55\times \frac{3^{6}}{4^{4}}
\Rightarrow k= 55

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Kuldeep Maurya

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