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.

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

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

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

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

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

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