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

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

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\(\Big(\frac{x}{4} - \frac{12}{x^2}\Big)^{12}\)

Tr+1 = (-1)r. \(^{12}C_r(\frac{x}{4})^{12-r}(\frac{12}{x^2})^r\)

Tr+1 = (-1)r. \(^{12}C_r(\frac{1}{4})^{12-r}(12)^r.(x)^{12-3r}\)

Term independent of x ⇒ 12 – 3r = 0 ⇒ r = 4

T5 = (-1)4.\(^{12}C_4(\frac{1}{4})^8(12)^4\) 

= \(\frac{3^6}{4^4}\).k

⇒ k = 55

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