Question

The power of $x$ in the term with the greatest coefficient in the expansion of ${\left( {1 - \dfrac{x}{2}} \right)^{10}}$ is
A. 2
B. 3
C. 4
D. 5
E. 6

Answer 1

Hint: In this question, we are given an expression ${\left( {1 - \dfrac{x}{2}} \right)^{10}}$. We have to find the highest power of $x$ with the greatest coefficient. Apply binomial distribution and expand the expression. Then, use combination and solve further to calculate the highest coefficient.

Formula used:
Binomial theorem –
${\left( {x + y} \right)^n} = \sum\limits_{r = 0}^n {{}^n{C_r}{x^{n - r}}{y^r}} $
Combination formula –
${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$, $n! = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times - - - - $

Complete step-by-step solution:
Given expression,
${\left( {1 - \dfrac{x}{2}} \right)^{10}}$
Using binomial theorem,
\[{\left( {1 - \dfrac{x}{2}} \right)^{10}} = {}^{10}{C_0}{\left( 1 \right)^{10}}{\left( {\dfrac{x}{2}} \right)^0} + {}^{10}{C_1}{\left( 1 \right)^9}{\left( {\dfrac{x}{2}} \right)^1} + {}^{10}{C_2}{\left( 1 \right)^8}{\left( {\dfrac{x}{2}} \right)^2} + {}^{10}{C_3}{\left( 1 \right)^7}{\left( {\dfrac{x}{2}} \right)^3} + {}^{10}{C_4}{\left( 1 \right)^6}{\left( {\dfrac{x}{2}} \right)^4} + {}^{10}{C_5}{\left( 1 \right)^5}{\left( {\dfrac{x}{2}} \right)^5} + - - - - - \]
\[ = 1 + \dfrac{{10!}}{{1!\left( {10 - 1} \right)!}}{\left( {\dfrac{x}{2}} \right)^1} + \dfrac{{10!}}{{2!\left( {10 - 2} \right)!}}\left( {\dfrac{{{x^2}}}{4}} \right) + \dfrac{{10!}}{{3!\left( {10 - 3} \right)!}}\left( {\dfrac{{{x^3}}}{8}} \right) + \dfrac{{10!}}{{4!\left( {10 - 4} \right)!}}\left( {\dfrac{{{x^4}}}{{16}}} \right) + \dfrac{{10!}}{{5!\left( {10 - 5} \right)!}}\left( {\dfrac{{{x^5}}}{{32}}} \right) + - - - - - \]
\[ = 1 + 5x + 11.25{x^2} + 15{x^3} + 13.13{x^4} + 7.88{x^5} + - - - - - - \]
Here, the highest coefficient is 15 in the given expression ${\left( {1 - \dfrac{x}{2}} \right)^{10}}$.
Therefore, the highest power is 3.

Hence, Option (B) is the correct answer i.e., 3.

Note: The key concept involved in solving this problem is the good knowledge of Binomial distribution. Students must remember that as the power increases, the expansion gets more difficult to compute. The Binomial Theorem can be used to easily calculate a binomial statement that has been raised to a very big power. The Binomial Theorem describes how to expand an expression raised to any finite power. A binomial theorem is a strong expansion technique that has applications in algebra, probability, and other fields.