First slide

Binomial theorem for positive integral Index

Question

In the expansion of ${(x^{3} - \frac{1}{x^{2}})}^{15}$ , the constant term, is

Moderate

A

$^{15} C_{9}$
B

0
C

$-^{15} C_{9}$
D

1

Solution

Let $(r + 1)^{th}$ term be the constant term in the expansion of ${(x^{3} - \frac{1}{x^{2}})}^{15}$
$∴ T_{r + 1} =^{15} C_{r} {(x^{3})}^{15 - r} {(- \frac{1}{x^{2}})}^{r}$ is independent of $x$
$\Rightarrow T_{r + 1} =^{15} C_{r} x^{45 - 5 r} (- 1)^{r}$ is independent of x
$\Rightarrow 45 - 5 r = 0 \Rightarrow r = 9$
Thus, tenth term is independent of $x$ and is given by
$T_{10} =^{15} C_{9} (- 1)^{9} = -^{15} C_{9}$

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