# JEE Binomial Theorem

This page will teach you how to master JEE binomial theorem. We highlight the main concepts, provide a list of examples with solutions, and include problems for you to try. Once you are confident, you can take the quiz to establish your mastery.

## JEE Conceptual Theory

As per JEE syllabus, the main concepts under binomial theorem are binomial theorem expansion, numerically greatest term in the binomial expansion, binomial coefficients, and binomial series.

**Binomial theorem expansion:**

Binomial theorem expansion for positive integral index: If \(n\) is a positive integer, then \[(x+y)^n=^nC_0x^ny^0+^nC_1 x^{n-1}y^1+^nC_2x^{n-2}y^2+\cdots+^nC_n x^0y^n.\]

General term in the expansion: General term in the expansion of \((x+y)^n\) is given by the \((r+1)^\text{th}\) term, i.e. \[T_{r+1}=^nC_rx^{n-r}y^r.\]

Binomial theorem expansion for any index: \((1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots.\)

**Numerically greatest term in the binomial expansion:**

Numerically greatest terms in the expansion of \((1+x)^n\) are \(T_p,T_{p+1}\) (values of both these terms are equal), if \(p\) is an integer, where \(p=\frac{(n+1)|x|}{|x|+1}.\)

Numerically greatest term in the expansion of \((1+x)^n\) is \(T_{c+1}\), if \(\frac{(n+1)|x|}{|x|+1}\) is not an integer, where \(c=\frac{(n+1)|x|}{|x|+1} .\)

**Binomial coefficients:**

Greatest binomial coefficient:

\(^nC_r\) is maximum at \(r= \begin{cases} \frac n2, &&\text{if } n \text{ is even} \\ \frac{n-1}{2}, \frac{n+1}{2}, &&\text{if } n \text{ is odd}. \end{cases} \)Properties of binomial coefficients:

\[\begin{align} ^nC_0+^nC_1+^nC_2+\cdots+^nC_n &= 2^n \\ ^nC_0-^nC_1+^nC_2-\cdots+(-1)^n \ ^nC_n &=0\\ ^nC_1-2 \cdot ^nC_2+3 \cdot ^nC_3-\cdots+(-1)^{n-1} n \cdot ^nC_n&=0 ~~(\text{for } n>1). \end{align}\]Use of differentiation and integration to find the sum of binomial coefficients

**Binomial series:**

Bino-arithmetic series:

\[a \ ^nC_0+(a+d) \ ^nC_1+(a+2d) \ ^nC_2+\cdots+(a+nd) \ ^nC_n=(2a+nd)2^{n-1}\]Bino-geometric series:

\[a \ ^nC_0+ab \ ^nC_1+ab^2 \ ^nC_2+\cdots+ab^n \ ^nC_n=a(1+b)^n\]Bino-harmonic series:

\[\frac{^nC_0}{a}+\frac{^nC_1}{a+d}+\frac{^nC_2}{a+2d}+\cdots+\frac{^nC_n}{a+nd}=\displaystyle \int_{0}^1 x^{a-1} \left(1+x^d \right)^ndx\]Bino-binomial series:

\[^nC_0 \ ^nC_n +^nC_1 \ ^nC_{n-1}+^nC_2 \ ^nC_{n-2}+\cdots+^nC_n \ ^nC_0=^{2n}C_n\]

## JEE Mains Problems

\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]

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\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]

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## JEE Advanced Problems

\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]

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Once you are confident of JEE Binomial Theorem, move on to JEE Multinomial Theorem.

**Cite as:**JEE Binomial Theorem .

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