First slide

Binomial theorem for positive integral Index

Question

The term independent of $x$ in the expansion of ${(x - \frac{3}{x^{2}})}^{18}$ is

Moderate

A

$^{18} C_{6}$
B

$^{18} C_{6} 3^{6}$
C

$^{18} C_{12}$
D

$3^{6}$

Solution

Given expansion ${(x - \frac{3}{x^{2}})}^{18}$
We know that $r^{t h}$ term in the expansion of ${(x + a)}^{n}$ is

$\begin{matrix} T_{r + 1} =^{n} C_{r} x^{n - r} a^{r} \end{matrix}$

then

$T_{r + 1} =^{18} C_{r} x^{18 - r} {(- \frac{3}{x^{2}})}^{r}$

$T_{r + 1} =^{18} C_{r} x^{18 - r} {(- 3)}^{r} {(x^{- 2})}^{r}$

$=^{18} C_{r} {(- 3)}^{r} x^{18 - 3 r} ................... (1)$

And this will be

Without $x$ only if $x^{18 - 3 r}$ compare with $x^{0}$ term independent of $x$ in the expansion

$\Rightarrow x^{18 - 3 r} = x^{0}$

$\Rightarrow 18 - 3 r = 0$

$∵ r = 6$ in Eqn (1)

We get

$\begin{matrix} ^{18} C_{6} {(- 3)}^{6} x^{18 - 18} \end{matrix}$

$\begin{matrix} ∵^{18} C_{6} 3^{6} \end{matrix}$

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