
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
164.4k+ views
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.
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.
Recently Updated Pages
Environmental Chemistry Chapter for JEE Main Chemistry

Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

Get P Block Elements for JEE Main 2025 with clear Explanations

Sets, Relations and Functions Chapter For JEE Main Maths

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Degree of Dissociation and Its Formula With Solved Example for JEE

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Instantaneous Velocity - Formula based Examples for JEE

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics
