Math · Algebra

Binomial Theorem formulas for JEE

Every Binomial Theorem formula you need for JEE, grouped by concept.

FormulasRevision notes
24 formulas3 concepts
01Binomial Theorem02Terms in Binomial Expansion03Properties of Binomial Coefficients
01

Binomial Theorem

7 formulas

Binomial Theorem for Any Index

(1+x)n=1+nx+n(n−1)2!x2+…∞(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \dots \infty

Infinite series expansion for fractional or negative indices.

applies when|x| < 1
binomial-theoreminfinite-seriesjee-advanced

Difference of Conjugate Binomials

(x+a)n−(x−a)n=2[nC1xn−1a+nC3xn−3a3+… ](x+a)^n - (x-a)^n = 2 [ {^nC_1} x^{n-1} a + {^nC_3} x^{n-3} a^3 + \dots ]

Subtracting (x-a)^n from (x+a)^n eliminates even powers of a.

applies whenn is a positive integer.
binomial-theoremconjugate

Binomial Theorem Expansion

(a+b)n=∑k=0nnCkan−kbk(a+b)^n = \sum_{k=0}^n {^nC_k} a^{n-k} b^k

Standard binomial expansion for positive integral indices.

applies whenn is a positive integer.
binomial-theoremexpansion

Binomial Expansion with Subtraction

(x−y)n=∑k=0n(−1)knCkxn−kyk(x-y)^n = \sum_{k=0}^n (-1)^k {^nC_k} x^{n-k} y^k

Expansion of (x-y)^n with alternating signs.

applies whenn is a positive integer.
binomial-theoremexpansionalternating-signs

Special Case (1-x)^n

(1−x)n=∑k=0n(−1)knCkxk(1-x)^n = \sum_{k=0}^n (-1)^k {^nC_k} x^k

Expansion of (1-x)^n with alternating signs.

applies whenn is a positive integer.
binomial-theoremexpansionspecial-case

Special Case (1+x)^n

(1+x)n=∑k=0nnCkxk(1+x)^n = \sum_{k=0}^n {^nC_k} x^k

Expansion of binomial with 1 and x.

applies whenn is a positive integer.
binomial-theoremexpansionspecial-case

Sum of Conjugate Binomials

(x+a)n+(x−a)n=2[nC0xn+nC2xn−2a2+… ](x+a)^n + (x-a)^n = 2 [ {^nC_0} x^n + {^nC_2} x^{n-2} a^2 + \dots ]

Summing (x+a)^n and (x-a)^n eliminates odd powers of a.

applies whenn is a positive integer.
binomial-theoremconjugate
02

Terms in Binomial Expansion

6 formulas

General Term

Tr+1=nCran−rbrT_{r+1} = {^nC_r} a^{n-r} b^r

The (r+1)-th term in the binomial expansion of (a+b)^n.

applies whenr is an integer, 0 <= r <= n.
general-termjee-advanced

Middle Term (Even n)

Tn2+1=nCn/2an/2bn/2T_{\frac{n}{2}+1} = {^nC_{n/2}} a^{n/2} b^{n/2}

The single middle term when n is even.

applies whenn is an even positive integer.
middle-termjee-advanced

Middle Terms (Odd n)

Tn+12 and Tn+32T_{\frac{n+1}{2}} \text{ and } T_{\frac{n+3}{2}}

The two middle terms when n is odd.

applies whenn is an odd positive integer.
middle-termjee-advanced

Numerically Greatest Term Locator

m=(n+1)∣x∣∣a∣+∣x∣m = \frac{(n+1)|x|}{|a| + |x|}

Calculation of m to find the numerically greatest term in (a+x)^n.

numerically-greatest-termjee-advanced

Number of Multinomial Terms

n+k−1Ck−1^{n+k-1}C_{k-1}

Total distinct terms in the expansion of a k-term multinomial raised to power n.

multinomial-theoremterm-countjee-advanced

Multinomial General Term

n!p1!p2!…pk!x1p1x2p2…xkpk\frac{n!}{p_1! p_2! \dots p_k!} x_1^{p_1} x_2^{p_2} \dots x_k^{p_k}

The general term in the expansion of a multinomial.

applies whenSum of non-negative integers p_1 + p_2 + ... + p_k = n
multinomial-theoremgeneral-termjee-advanced
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03

Properties of Binomial Coefficients

11 formulas

Coefficient Symmetry

nCr=nCn−r^nC_r = {^nC_{n-r}}

Binomial coefficients are symmetric.

symmetryjee-advanced

Binomial Coefficient

nCr=n!r!(n−r)!^nC_r = \frac{n!}{r!(n-r)!}

Definition of the binomial coefficient nCr.

applies when0≤r≤n0 \le r \le n, where n is a non-negative integer.
combinationscoefficients

Sum of Coefficients with Powers

∑r=0nnCrxr=(1+x)n\sum_{r=0}^n {^nC_r} x^r = (1+x)^n

Generalized sum identity extracted from Exercise 7.1 Q14.

applies whenn is a positive integer.
coefficientsseries-sumexercise-derived

Pascal's Identity

nCr+nCr−1=n+1Cr^nC_r + ^nC_{r-1} = ^{n+1}C_r

Addition rule for adjacent binomial coefficients.

pascal-triangleidentities

Coefficient Extraction Property

r⋅nCr=n⋅n−1Cr−1r \cdot {^nC_r} = n \cdot {^{n-1}C_{r-1}}

Derivative-based property to reduce the index and upper value.

applies whenr > 0
series-sumderivativesjee-advanced

Coefficient Integration Property

nCrr+1=n+1Cr+1n+1\frac{^nC_r}{r+1} = \frac{^{n+1}C_{r+1}}{n+1}

Integral-based property to increase the index and upper value.

series-sumintegrationjee-advanced

Sum of Binomial Coefficients

∑r=0nnCr=2n\sum_{r=0}^n {^nC_r} = 2^n

The sum of all coefficients in the expansion of (1+x)^n.

applies whenn is a positive integer.
coefficientsseries-sum

Alternating Sum of Coefficients

∑r=0n(−1)rnCr=0\sum_{r=0}^n (-1)^r {^nC_r} = 0

The alternating sum of coefficients evaluates to 0.

applies whenn is a positive integer.
coefficientsseries-sumalternating

Sum of Even/Odd Positioned Coefficients

nC0+nC2+⋯=nC1+nC3+⋯=2n−1^nC_0 + ^nC_2 + \dots = ^nC_1 + ^nC_3 + \dots = 2^{n-1}

The sum of coefficients at entirely even or entirely odd positions.

applies whenn > 0
series-sumparityjee-advanced

Sum of Product of Coefficients

∑r=0n−knCr⋅nCr+k=2nCn−k\sum_{r=0}^{n-k} {^nC_r} \cdot {^nC_{r+k}} = {^{2n}C_{n-k}}

The sum of products of coefficients offset by k.

applies whenk is a non-negative integer, k <= n
series-sumproductsjee-advanced

Sum of Squares of Coefficients

∑r=0n(nCr)2=2nCn\sum_{r=0}^n (^nC_r)^2 = {^{2n}C_n}

The sum of the squares of all binomial coefficients for a given n.

series-sumsquaresjee-advanced
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