Question

The term void of $x$ in the expansion of ${\left( {x - \dfrac{3}{{{x^2}}}} \right)^{18}}$ is:
A.$^{18}{C_6}$
B.$^{18}{C_6}{3^6}$
C.$^{18}{C_5}$
D.$^{18}{C_6}{3^{12}}$

Answer 1

Hint: We want to find a term void of $x$ which also means a term independent of $x$. Hence, find the term in the expansion where power of $x$ is zero. Use the formula of the general term of binomial expansion and find the power of $x$. Equate it 0 to find the value of $r$, substitute the value of $r$ in the formula of the general term to find the term void of $x$.

Complete step-by-step answer:
We have to find the term in the binomial expansion of ${\left( {x - \dfrac{3}{{{x^2}}}} \right)^{18}}$ that is void of $x$ or we can say, independent of $x$.
Thus, we need to find the term in the expansion such that the power of $x$ is 0.
First of all, let us find out the general term in the binomial expansion of ${\left( {x - \dfrac{3}{{{x^2}}}} \right)^{18}}$.
The formula of general term is, ${T_{r + 1}}{ = ^n}{C_r}{a^{n - r}}{b^r}$, where $0 \leqslant r < n$ in the expansion of ${\left( {a + b} \right)^n}$.
On substituting $a = x,b = - \dfrac{3}{{{x^2}}},n = 18$ in the general term formula we get,
${T_{r + 1}}{ = ^{18}}{C_r}{\left( x \right)^{18 - r}}{\left( { - \dfrac{3}{{{x^2}}}} \right)^r}$
Combine the terms of $x$.
$
  {T_{r + 1}}{ = ^{18}}{C_r}{\left( x \right)^{18 - r}}{\left( { - \dfrac{3}{{{x^2}}}} \right)^r} \\
  {T_{r + 1}}{ = ^{18}}{C_r}{\left( x \right)^{18 - r}}{\left( { - 3} \right)^r}{\left( x \right)^{ - 2r}} \\
  {T_{r + 1}}{ = ^{18}}{C_r}{\left( { - 3} \right)^r}{\left( x \right)^{18 - 3r}} \\
 $
Now, to find the term independent of $x$, so put $18 - 3r = 0$ to find the value of $r$.
$
  18 - 3r = 0 \\
  18 = 3r \\
  r = 6 \\
 $
On substituting the value of $r$in general term formula ${T_{r + 1}}{ = ^{18}}{C_r}{\left( { - 3} \right)^r}{\left( x \right)^{18 - 3r}}$, we get
$
  {T_{6 + 1}}{ = ^{18}}{C_6}{\left( { - 3} \right)^6}{\left( x \right)^{18 - 3\left( 6 \right)}} \\
  {T_7}{ = ^{18}}{C_6}{\left( { - 3} \right)^6} \\
  {T_7}{ = ^{18}}{C_6}{\left( 3 \right)^6} \\
 $
Thus, the term void of $x$ is $^{18}{C_6}{3^6}$.
Hence, B is the correct option.

Note: Any number raised to the power 0 is 1. Hence, to get a term independent of $x$ find the term in which the power of $x$ is zero. Many students do this question by expanding the expression term by term, which makes the solution unnecessarily long.