Find the coefficient of $x^{6}$ and the coefficient of $x^{2}$ in ${(x^{2} - \frac{1}{x^{3}})}^{6}$

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Solution

${(x^{2} - \frac{1}{x^{3}})}^{6} =^{6} C_{r} (x^{2})^{6 - r} {(\frac{1}{x^{3}})}^{r}$
$12 - 2 r - 3 r = 6$ $12 - 5 r = 2$
$5 r = 6$ $10 = 5 r$
$r = 1.5$ Not possible $r = 2$
$^{6} C_{2} = \frac{6 \times 5 \times 4!}{4! \times 2}$
$= 15$

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Find the coefficient of x6 and the coefficient of x2 in (x2−1x3)6

(x2−1x3)6=6Cr(x2)6−r(1x3)r12−2r−3r=6 12−5r=25r=6 10=5rr=1.5 Not possible r=2 6C2=6×5×4!4!×2 =15

Find the coefficient of $x^{6}$ and the coefficient of $x^{2}$ in ${(x^{2} - \frac{1}{x^{3}})}^{6}$

${(x^{2} - \frac{1}{x^{3}})}^{6} =^{6} C_{r} (x^{2})^{6 - r} {(\frac{1}{x^{3}})}^{r}$
$12 - 2 r - 3 r = 6$ $12 - 5 r = 2$
$5 r = 6$ $10 = 5 r$
$r = 1.5$ Not possible $r = 2$
$^{6} C_{2} = \frac{6 \times 5 \times 4!}{4! \times 2}$
$= 15$