Imagine you’re sharing a pack of candies with your friends and trying to figure out how many different ways you can choose who gets what. That simple idea of counting choices is what binomial coefficients are all about. In this article on Greatest Binomial Coefficient, we’ll explain what it means in the easiest way possible, show you how to find the biggest term in a binomial expansion in mathematics, and walk you through the key formulas you’ll need for exams.
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A binomial expression is an algebraic expression that contains exactly two terms. These terms may involve constants, variables, rational powers, or irrational expressions. Some common examples include:
$(a + b)^2$
$\left(\sqrt{x} + \frac{k}{x^2}\right)^5$
$(x + 9y)^{-2/3}$
Binomials form the foundation of the Binomial Theorem and are used to generate expansions, coefficients, and individual terms.
The Binomial Theorem provides a structured way to expand $(a + b)^n$ without manual multiplication.
For any positive integer $n$, the expansion is:
$(a + b)^n = \binom{n}{0} a^n + \binom{n}{1} a^{n - 1} b + \binom{n}{2} a^{n - 2} b^2 + \dots + \binom{n}{n - 1} a b^{n - 1} + \binom{n}{n} b^n$
Each term in the expansion is influenced by:
The binomial coefficient
The powers of $a$ and $b$
Their relative magnitudes
This structure allows us to determine which term becomes the numerically greatest term.
A binomial coefficient represented by $\binom{n}{r}$ or $^nC_r$ indicates the number of ways to choose $r$ items from $n$ items.
It is defined as:
$\binom{n}{r} = \frac{n!}{r!(n-r)!}$
Binomial coefficients increase from $r = 0$ to about the middle term.
Coefficients then decrease symmetrically.
The coefficient with the maximum value influences the largest term in the expansion.
In the expansion of $(x + a)^n$, each term is influenced by the binomial coefficient and the powers of $x$ and $a$.
The numerically greatest term is the one with the highest absolute value among all expanded terms.
The numerically greatest term in $(x + a)^n$ is the term whose magnitude is larger than all other terms in that expansion. This depends on:
Binomial coefficients
The ratio $\left|\frac{x}{a}\right|$
The power $n$
The approximate term where the greatest value occurs is found using:
$r = \frac{n + 1}{1 + \left|\frac{x}{a}\right|}$
The candidate greatest terms are $T_r$ and $T_{r + 1}$.
This method allows you to identify the term in the expansion that has the maximum value without expanding the entire expression.
1. Compute the value of $r$
Use:
$r = \frac{n + 1}{1 + \left|\frac{x}{a}\right|}$
This estimate tells you where the peak occurs among the binomial terms.
2. Check whether $r$ is an integer
Here $[r]$ denotes the integral part (or floor value) of $r$.
This technique is especially valuable because:
You avoid full binomial expansion.
It is used widely in JEE Advanced, JEE Main, CUET, Olympiad problems, and Class 11 Algebra.
It provides direct insight into the behavior of binomial expressions and coefficients.
Understanding the greatest binomial coefficient helps you evaluate maximum term problems quickly and accurately.
The greatest binomial coefficient refers to the maximum value among all binomial coefficients in the expansion of $(a+b)^n$. This concept plays a key role in binomial theorem problems, especially in Class 11 and 12 mathematics, JEE, CUET, and other competitive exams.
In the expansion $(1+x)^n = {}^nC_0 + {}^nC_1 x + {}^nC_2 x^2 + \cdots + {}^nC_n x^n$, the coefficients increase up to a certain point and then decrease. The largest among them is called the greatest binomial coefficient.
The greatest binomial coefficient is the largest value of ${}^nC_r$ for a given $n$.
A key property used is symmetry: ${}^nC_r = {}^nC_{n-r}$
Because of this symmetry, the greatest coefficient lies near the middle term of the binomial expansion.
Example:
In $(1+x)^6$, the coefficients are $1, 6, 15, 20, 15, 6, 1$
The greatest binomial coefficient is ${}^6C_3 = 20$
The position of the greatest binomial coefficient depends on whether $n$ is even or odd.
General observation:
Coefficients increase as $r$ approaches $\dfrac{n}{2}$
Coefficients decrease after crossing $\dfrac{n}{2}$
This makes the middle term (or middle terms) the most important when finding the greatest coefficient.
If $n = 2m$, there is only one greatest binomial coefficient.
It occurs at $r = m$
Greatest binomial coefficient: ${}^{2m}C_m$
Example: For $(1+x)^8$,
Greatest coefficient: ${}^8C_4 = 70$
If $n = 2m + 1$, there are two equal greatest binomial coefficients.
They occur at $r = m$ and $r = m+1$
Greatest binomial coefficients: ${}^{2m+1}C_m = {}^{2m+1}C_{m+1}$
Example: For $(1+x)^7$,
${}^7C_3 = {}^7C_4 = 35$
Both are the greatest coefficients.
For the expansion of $(1+x)^n$:
If $n$ is even: Greatest coefficient $= {}^nC_{n/2}$
If $n$ is odd: Greatest coefficients $= {}^nC_{(n-1)/2}$ and ${}^nC_{(n+1)/2}$
This formula is frequently used in objective questions and short-answer problems.
The greatest binomial coefficient depends only on $n$, while the greatest term depends on both $n$ and the value of $x$ in $(1+x)^n$ or $(a+bx)^n$.
This distinction is important in numerical maximum problems.
We have given below the important differences between the greatest term and greatest coefficients, to help you understand and identify each of them:
| Aspect | Greatest Binomial Coefficient | Greatest Term |
|---|---|---|
| Depends on | Only $n$ | $n$ and $x$ |
| Involves variable | No | Yes |
| Number of terms | One or two | One or two |
| Common exam use | Conceptual MCQs | Numerical value problems |
The greatest term also has the greatest binomial coefficient when $x = 1$ in $(1+x)^n$.
Example:
In $(1+1)^n = 2^n$, the numerically greatest term corresponds to the greatest coefficient because all powers of $x$ are equal.
For values of $x \ne 1$, the greatest term may shift away from the greatest coefficient.
Greatest binomial coefficient lies at the middle
Even $n$ → one greatest coefficient
Odd $n$ → two greatest coefficients
Greatest term and greatest coefficient are not always the same
Example 1:
The greatest value of the term independent of $x$ in the expansion of
$\left(x \sin a + x^{-1} \cos a\right)^{10}$ is
$2^5$
$\dfrac{10!}{(5!)^2}$
$\dfrac{1}{2^5} \cdot \dfrac{10!}{(5!)^2}$
none of these
Solution
$T_{r+1} = {}^{10}C_r (x \sin a)^{10-r} \left(\dfrac{\cos a}{x}\right)^r$
The term is independent of $x$ when
$10 - r - r = 0 \Rightarrow r = 5$
The term independent of $x$ is
${}^{10}C_5 \sin^5 a \cos^5 a$
${}^{10}C_5 \sin^5 a \cos^5 a = {}^{10}C_5 \cdot \dfrac{1}{2^5} (\sin 2a)^5$
Since $(\sin 2a)^5 \le 1$,
${}^{10}C_5 \cdot \dfrac{1}{2^5} (\sin 2a)^5 \le {}^{10}C_5 \cdot \dfrac{1}{2^5}$
Hence, the correct answer is option (3).
Example 2:
If $\ln n$ is an even positive integer, then the condition that the greatest term in the expansion of $(1+x)^n$ also has the greatest coefficient $(x>0)$ is
$\dfrac{n}{n+2} < x < \dfrac{n+2}{n}$
$\dfrac{n+1}{n} < x < \dfrac{n}{n+1}$
$\dfrac{n}{n+4} < x < \dfrac{n+4}{4}$
none of these
Solution
Let
$n = 2m$
The greatest binomial coefficient is
${}^nC_{n/2} = {}^{2m}C_m$
The corresponding term is
$T_{m+1}$
For $T_{m+1}$ to be the greatest term,
$m < \dfrac{2m+1}{1+\left|\dfrac{1}{x}\right|} < m+1$
Solving and substituting $m = \dfrac{n}{2}$,
$\dfrac{n}{n+2} < x < \dfrac{n+2}{n}$
Hence, the correct answer is option (1).
Example 3:
If for some positive integer $n$, the coefficients of three consecutive terms in the expansion of $(1+x)^{n+5}$ are in the ratio $5:10:14$, then the largest coefficient is
$462$
$330$
$792$
$252$
Solution
Let
$n+5 = N$
${}^NC_{r-1} : {}^NC_r : {}^NC_{r+1} = 5 : 10 : 14$
$\dfrac{{}^NC_r}{{}^NC_{r-1}} = \dfrac{N+1-r}{r} = 2$
$\dfrac{{}^NC_{r+1}}{{}^NC_r} = \dfrac{N-r}{r+1} = \dfrac{7}{5}$
Solving,
$r = 4$ and $N = 11$
Thus the expansion is $(1+x)^{11}$
The largest coefficient is
${}^{11}C_6 = 462$
Hence, the correct answer is option (1).
Example 4:
Find the numerically greatest term in the expansion of $(2+3x)^9$ when $x = \dfrac{2}{3}$
$6^{\text{th}}$ term
$5^{\text{th}}$ term
$5^{\text{th}}$ and $6^{\text{th}}$ terms
$8^{\text{th}}$ term
Solution
Here,
$a = 2$ and $b = 3x = 2$
$m = \dfrac{n+1}{1+\left|\dfrac{a}{b}\right|} = \dfrac{10}{1+1} = 5$
Since $m$ is an integer, there are two numerically greatest terms:
$T_5$ and $T_6$
Hence, the correct answer is option (3).
Example 5:
For the $9^{\text{th}}$ term in the binomial expansion of $(3+6x)^n$ (in increasing powers of $6x$), when $x = \dfrac{3}{2}$, find $k + n_0$
$24$
$15$
$17$
$20$
Solution
$\dfrac{n+1}{1+\left|\dfrac{a}{b}\right|} = \dfrac{n+1}{1+\left|\dfrac{3}{6 \cdot \frac{3}{2}}\right|} = \dfrac{3(n+1)}{4}$
Since the $9^{\text{th}}$ term is the greatest,
$8 < \dfrac{3(n+1)}{4} < 9$
$32 < 3(n+1) < 36$
$10.66 < n+1 < 12$
$9.66 < n < 11$
Therefore,
$n_0 = 10$
$k = \dfrac{{}^{10}C_6 \cdot 3^4 \cdot 6^6}{{}^{10}C_3 \cdot 3^7 \cdot 6^3} = 14$
$k + n_0 = 24$
Hence, the correct answer is 24.
This section gives you a quick overview of all the important subtopics connected to the Binomial Theorem so you know exactly what to study and how each idea fits into exam preparation.
Binomial Theorem for any Index
Binomial Inside Binomial
Important Results of Binomial Theorem for any Index
Last Digits and Remainder using the Binomial Expansion
This section brings together all the essential NCERT study materials you need to build a strong foundation in the Binomial Theorem. You’ll find chapter notes, solved examples, and exemplar solutions that make understanding concepts easier and support exam-focused learning.
NCERT Maths Class 11 Notes for Chapter 8 - Binomial Theorem and its applications
NCERT Maths Class 11 Solutions for Chapter 8 - Binomial Theorem and its applications
NCERT Maths Class 11 Exemplar Solutions for Chapter 8 - Binomial Theorem and its applications
This section gives you a set of exam-oriented practice questions that help you understand how to identify the greatest binomial coefficient and the largest term in a binomial expansion. It’s perfect for strengthening your skills for JEE, CUET, and board exams.
Greatest Term Numerically- Practice Question MCQ
We have shared below the links to practice questions for the topics related to binomial theorem:
Frequently Asked Questions (FAQs)
The greatest binomial coefficient refers only to the coefficient $\binom{n}{r}$.
The greatest term refers to the full expression $T_r = \binom{n}{r}a^{n-r}x^r$, which includes powers of $a$ and $x$. So a smaller coefficient might produce a larger term if multiplied by large variable values.
Yes, when both $a$ and $b$ are equal in magnitude (like $(1 + 1)^n$ or $(x + x)^n$), the greatest coefficient appears at the middle term(s), such as $\binom{n}{n/2}$. But if $x$ and $a$ differ in size, the greatest term may shift away from the centre.
Because each term in the expansion has the form $\binom{n}{r}a^{n-r}x^r$, the relative sizes of $x$ and $a$ directly influence which term becomes largest. The ratio helps balance the falling power of $a$ and the rising power of $x$.
To locate the numerically greatest term, compute $r = \frac{n + 1}{1 + |\frac{x}{a}|}$. If $r$ is an integer, both $T_r$ and $T_{r+1}$ are equal and greatest; if not, $T_{[r]+1}$ is the numerically greatest term.
The greatest binomial coefficient is the largest value among all coefficients in the expansion of $(a + b)^n$. Since the coefficients rise and then fall symmetrically, the maximum value usually appears at the middle term(s), such as $\binom{n}{\lfloor n/2 \rfloor}$.
