Fin (a) the coefficient of $x^{9}$ , (b) the term independent of x in the expansion of $(x^{2} - 1 / 3 x)^{9}$ .

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Solution

The term of $x^{9}$ in ${(x^{2} - \frac{1}{3 x})}^{9}$ will occur in 4th term i.e., r = 3 and its coefficient will be
$^{9} C_{3} (x^{2})^{6} {(- \frac{1}{3 x})}^{3} = - \frac{9!}{6! \cdot 3!} \cdot \frac{1}{3^{3}}$
$= - \frac{9 \times 8 \times 7}{3 \cdot 2 \cdot 1} \times \frac{1}{27} = - \frac{28}{9} .$
Similarly term independent of x will be 7th i.e., r = 6 and $T_{7} = \frac{28}{243}$ .

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