If $\left( {{{{3^6}} \over {{4^4}}}} \right)k$ is the term, independent of x, in the binomial expansion of ${\left( {{x \over 4} - {{12} \over {{x^2}}}} \right)^{12}}$, then k is equal to ___________.
Answer (integer)
55
Solution
${\left( {{x \over 4} - {{12} \over {{x^2}}}} \right)^{12}}$<br><br>$${T_{r + 1}} = {( - 1)^r}\,.\,{}^{12}{C_r}{\left( {{x \over 4}} \right)^{12 - r}}{\left( {{{12} \over {{x^2}}}} \right)^r}$$<br><br>$${T_{r + 1}} = {( - 1)^r}\,.\,{}^{12}{C_r}{\left( {{1 \over 4}} \right)^{12 - r}}{\left( {12} \right)^r}\,.\,{(x)^{12 - 3r}}$$<br><br>Term independent of x $\Rightarrow$ 12 $-$ 3r = 0 $\Rightarrow$ r = 4<br><br>$${T_5} = {( - 1)^r}\,.\,{}^{12}{C_r}{\left( {{1 \over 4}} \right)^8}{\left( {12} \right)^4} = {{{3^6}} \over {{4^4}}}.\,k$$<br><br>$\Rightarrow$ k = 55
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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