Greatest Binomial Coefficient

Understanding the Greatest Binomial Coefficient

The greatest binomial coefficient plays a key role in identifying the numerically largest term in a binomial expansion. Whether you are preparing for JEE, CUET, Class 11–12 exams, or competitive mathematics, understanding binomial expressions, binomial coefficients, and the logic behind the greatest term helps you solve a wide range of algebra questions confidently. Below is a clear, structured explanation of the concepts involved.

What is a Binomial Expression?

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.

Binomial Theorem

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.

Binomial Coefficient: Definition and Formula

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:

$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Key Properties for Greatest Term Problems

• 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.

Numerically Greatest Value in a Binomial 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.

Definition

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 $n$

General Formula for the Greatest Term Index

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 $T_{r + 1}$.

Method to Find the Numerically Greatest Term of $(x + a)^n$

This method allows you to identify the term in the expansion that has the maximum value without expanding the entire expression.

Step-by-Step Procedure

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

• If $r$ is an integer:
  Both $T_r$ and $T_{r + 1}$ have equal numerical values and are the greatest terms in the expansion.
• If $r$ is not an integer:
  The term $T_{[r] + 1}$ (i.e., the term after the integer part of $r$) is the numerically greatest term.

Here $[r]$ denotes the integral part (or floor value) of $r$.

Why This Method Matters

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.

Solved Examples Based on Greatest Binomial Coefficient:

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
1) $2^5$
2) $\frac{10!}{(5!)^2}$
3) $\frac{1}{2^5} \cdot \frac{10!}{(5!)^2}$
4) none of these

Solution

$
T_{r+1}={ }^{10} C_r(x \sin \alpha)^{10-r} \cdot\left(\frac{\cos \alpha}{x}\right)^r
$

It is independent of $x$ if $r=5$.
The term independent of $x={ }^{10} \mathrm{C}_5 \cdot \sin ^5 \mathrm{a} \cdot \cos ^5 \mathrm{a}$

$
={ }^{10} C_5 \cdot \frac{1}{2^5}(\sin 2 \alpha)^5 \leq{ }^{10} C_5 \cdot \frac{1}{2^5}
$

Hence, the answer is the option 3.

Example 2: $\ln n$ is an even positive integer, then the condition that the greatest term in the expansion of $(1+x)^n$ many have the greatest coefficient also, is ( x is positive)
1) $\frac{n}{n+2}<x<\frac{n+2}{n}$
2) $\frac{n+1}{n}<x<\frac{n}{n+1}$
3) $\frac{n}{n+4}<x<\frac{n+4}{4}$
4) none of these

Solution

Let $\mathrm{n}=2 \mathrm{~m}$
If $n$ is even then the greatest binomial coefficient $={ }^n C_{n / 2}={ }^{2 m} C_m$

$
=(m+1) \text { th term }=T_{m+1}
$
Now, since $T_{m+1}$ is the greatest term

$
m<\frac{(2 m+1)}{1+\left|\frac{1}{x}\right|}<(m+1)
$

Solving it and putting $m=n / 2$ we get

$
\frac{n}{n+2}<x<\frac{n+2}{n}
$
Hence, the answer is the option 1.

Example 3: If for some positive integer n , the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$, then the largest coefficient in this expansion is.
1) $462$
2) $330$
3) $792$
4) $252$

Solution

$
\begin{aligned}
& \text { Let } \mathrm{n}+5=\mathrm{N} \\
& { }^{\mathrm{N}} \mathrm{C}_{\mathrm{r}-1}:{ }^{\mathrm{N}} \mathrm{C}_{\mathrm{r}}:{ }^{\mathrm{N}} \mathrm{C}_{\mathrm{r}+1}=5: 10: 14 \\
& \Rightarrow \frac{{ }^{\mathrm{N}} \mathrm{C}_{\mathrm{r}}}{{ }^{\mathrm{N}_{\mathrm{C}}} \mathrm{C}_{\mathrm{r}-1}}=\frac{\mathrm{N}+1-\mathrm{r}}{\mathrm{r}}=2 \\
& \mathrm{~N}_{\mathrm{C}_{\mathrm{r}+1}}^{\mathrm{N}_{\mathrm{C}_{\mathrm{r}}}}=\frac{\mathrm{N}-\mathrm{r}}{\mathrm{r}+1}=\frac{7}{5} \\
& \Rightarrow \mathrm{r}=4, \mathrm{~N}=11 \\
& \Rightarrow(1+\mathrm{x})^{11}
\end{aligned}
$

Largest coefficient $={ }^{11} \mathrm{C}_6=462$
Hence, the answer is option (1).

Example 4: Find the numerically greatest term in the expansion of $(2+3 x)^9$, when $x=\frac{2}{3}$
1) $6^{\text {th }}$ term
2) $5^{\text {th }}$ term
3) $5^{\text {th }}$ term and $6^{\text {th }}$ term
4) $8^{\text {th }}$ term

Solution
Here $a=2$ and $b=3 x=2($ As $x=2 / 3)$

So,

$
m=\frac{n+1}{1+|a / b|}=\frac{10}{1+1}=5
$

As $m$ is an integer, so there are two numerically greatest terms
$T_m$ and $T_{m+1}: T_5$ and $T_6$
Hence, the answer is the option 3.
Example 5: Let for the $9^{\text {th }}$ term in the binomial expansion of $(3+6 x)^{\mathrm{n}}$, in the increasing powers of 6 x , $x=\frac{3}{2}$
$k+n_0$ is equal to :
1) $24$
2) $15$
3) $17$
4) $20$

Solution

$
\frac{\mathrm{n}+1}{1+\left|\frac{\mathrm{a}}{\mathrm{b}}\right|}=\frac{\mathrm{n}+1}{1+\left|\frac{3}{6 \times \frac{3}{2}}\right|}=\frac{3(\mathrm{n}+1)}{4}
$
As $g^{\text {th }}$ term is greatest,

$
\begin{aligned}
& \therefore 8<\frac{3(\mathrm{n}+1)}{4}<9 \\
& 32<3(\mathrm{n}+1)<36 \\
& 10.66<\mathrm{n}+1<12 \\
& \quad 9.66<\mathrm{n}<11 \\
& \therefore \mathrm{n}_0=10 \\
& \mathrm{k}=\frac{{ }^{10} \mathrm{C}_6 \cdot(3)^4(6)^6}{{ }^{10} \mathrm{C}_3 \cdot 3^7(6)^3}=14 \\
& \therefore \quad \mathrm{k}+\mathrm{n}_0=24
\end{aligned}
$
Hence, the answer is $24$ .

Last Digits and Remainder using the Binomial Expansion