Careers360 Logo
34 Percentile in JEE Mains 2025 Means How Many Marks? - Check Details

Binomial Theorem - Formula, Expansion, Problems and Applications - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Binomial Theorem and Expression of Binomial Theorem is considered one the most difficult concept.

  • Properties of Binomial Theorem and Binomial Coefficient (Part 1), Properties of Binomial Theorem and Binomial Coefficient (Part 2) is considered one of the most asked concept.

  • 150 Questions around this concept.

Solve by difficulty

Which of the following is NOT a binomial expression?

For natural numbers m,n if (1-y)^{m}(1+y)^{n}=1+a_{1}y+a_{2}y^{2}+........,\; and\; a_{1}=a_{2}=10,then\; (m,n)\; is :

Which of the following is NOT a Binomial Expression?

Which one of the following is not a binomial expression?

Which of the following can NOT be expressed as a binomial theorem?

The expression [x+(x31)1/2]5+[x(x31)1/2]5 is a polynomial of degree

(x+2)7=7C0×27+7C1×26x+7C2×25x2+7C3×24x3+7C4×23x4+7C5×22x5+7C6×2x6+7C7×x7 

Binomial coefficient of x5  is  

Amrita Vishwa Vidyapeetham | B.Tech Admissions 2025

Recognized as Institute of Eminence by Govt. of India | NAAC ‘A++’ Grade | Upto 75% Scholarships

UPES B.Tech Admissions 2025

Ranked #42 among Engineering colleges in India by NIRF | Highest CTC 50 LPA , 100% Placements | Last Date to Apply: 25th Feb

Which of the following is Binomial theorem ? 

Which of the following statement is wrong ? 

JEE Main 2025 College Predictor
Know your college admission chances in NITs, IIITs and CFTIs, many States/ Institutes based on your JEE Main result by using JEE Main 2025 College Predictor.
Try Now

Which of the statement is incorrect ? 

Concepts Covered - 3

Binomial Theorem and Expression of Binomial Theorem

An algebraic expression consisting of only two terms is called a Binomial Expression.

eg(a+b)2,(x+kx2)5,(x+9y)2/3
If we wanted to expand (x+y)52, we might multiply (x+y) by itself fifty-two times. This could take hours!
But if we examine some simple expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.

(x+y)2=x2+2xy+y2(x+y)3=x3+3x2y+3xy2+y3(x+y)4=x4+4x3y+6x2y2+4xy3+y4
On examining the exponents, we find that with each successive term, the exponent for x decreases by 1 and the exponent for y increases by 1. The sum of the two exponents is n for each term.
Also the coefficients for (x+y)n are equal to (n0),(n1),(n2),,(nn)
where, (nr)=C(n,r)=nCr=n!r!(nr)!
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial expression.
For any natural number n, binomial expansion is

(x+y)n=nC0xn+nC1xn1y+nC2xn2y2++nCnyn where, nN=r=0n(nCr)xnryr
The combination (nr) or nCr is called a binomial coefficient.

Properties of Binomial Theorem and Binomial Coefficient (Part 1)

Properties of Binomial Theorem and Binomial Coefficient

  1. 1. If n is a natural number, then the expansion (x+y)n has n+1 terms.
    2. In each term, the sum of the index of ' x ' and ' y ' is equal to ' n '.
    3. The sum of two consecutive binomial coefficients,
    4. nCr=nCnr=n!r!(nr)!

    So, if nCx=nCy, then either x=y or n=x+y

Properties of Binomial Theorem and Binomial Coefficient (Part 2)

Properties of Binomial Theorem and Binomial Coefficient

5. Binomial coefficients of the term equidistant from the beginning and end are equal.

Eg, second binomial coefficient from start =nC1, and second binomial coefficient from the end =nCn1

These are equal as 1+(n1)=n
6. nCr=nrn1Cr1

nCr=n!r!(nr)!=n(n1)!r(r1)!(nr)!=nrn1Cr1

7. nCrnCr1=nr+1r

Study it with Videos

Binomial Theorem and Expression of Binomial Theorem
Properties of Binomial Theorem and Binomial Coefficient (Part 1)
Properties of Binomial Theorem and Binomial Coefficient (Part 2)

"Stay in the loop. Receive exam news, study resources, and expert advice!"

Get Answer to all your questions

Back to top