# BINOMIAL THEOREM

### INTRODUCTION

• An expression containing two terms connected by + or – sign is called a BINOMIAL. For example, a + b, , a – y2, 2b – 3c etc. are binomial expressions.
• Similarly, an expression containing three terms is called a TRINOMIAL. In general, expressions containing more than two terms are called MULTINOMIALS.
• The general form of the binomial expression is x + a and the expansion of (x + a)n, n being a positive integer, is called the BINOMIAL THEOREM. Sir Isaac Newton first gave this theorem.

### BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX

Statement : If , then PROPERTIES OF THE EXPANSION OF (x + a)n
1. The number of terms in the expansion of (x + a)n is n + 1, i.e., one more than the index n.
2. The sum of the powers of x and a in each term is n, thus it is a homogeneous expansion.
3. The (r + 1)th term in the expansion is nCr xn – r ar and is called the general term, i.e., Tr + 1 = nCr xn – r ar.
4. nC0, nC1, nC2 ......, nCn are coefficients of the successive terms and are called the binomial coefficients.
5. The binomial coefficients of terms equidistant from the beginning and end are same as nCr = nCn – r
6. The binomial expansion is briefly written as .
7. Putting –a for a, we have The terms in this expansion are alternatively positive and negative and the last term is positive or negative according as n is even or odd.
1. Putting x = 1 and a = x in the expansion of (x + a)n, we have This is expansion of (1 + x)n is ascending powers of x.
1. Putting a = 1 in the expansion of (x + a)n, we have This is expansion of (1 + x)n is descending powers of x. The coefficient (r + 1)th term and the coefficient of xr in (1 + x)n is  or , both being equal.
1. The kth term from the end in the expansion of (x + a)n is
(n – k + 2)th term from the beginning.
2. We note that = 2 [sum of  the terms at even places] = 2 [sum of the term at odd places]

MIDDLE TERM OF TERMS IN THE EXPANSION OF (x + a)n
The number of terms in the expansion of (x + a)n is n + 1. Therefore,
If n is even then there is only one middle term, viz. th term.
If n is odd then there are two middle terms, viz. th and th terms.

GREATEST COEFFICIENT IN THE EXPANSION OF (x + a)n
The coefficients are where the coefficient of the general term is nCr. We have to find the value of r for which nCr has the greatest value. We know that if n is even nCr is greatest when and if n is odd nCr is greatest for or .
Hence, if n is even the greatest coefficient is , and if n is odd, the greatest coefficient is nC(n – 1)/2 or nC(n + 1)/2 both being equal.

NUMERICALLY GREATEST TERM IN THE EXPANSION OF (x + a)n
Let Tr and Tr + 1 be rth and (r + 1)th terms respectively in the binomial expansion of (x + a)n. Assume that x and a are positive
Then Tr =  nCr – 1 xn – r + 1 ar – 1 and Tr +  1= nCr xn – r ar or Hence, According as i.e., according as i.e., according as i.e., according as i.e., according as ……….(I)
Now the value of may be an integer or fraction. Therefore, two cases arise.

Case I : When is an integer, say, p
From (I), Tr + 1 > Tr if r < p, otherwise Tr + 1 Tr
∴ For  r = 1, 2, ...., p – 1 we have Tr + 1 > Tr,
For r = p we have Tr + 1 = Tr
And for r = p + 1, p + 2, ..., we have Tr + 1 < Tr.
∴ Hence in this case Tp = Tp + 1 and these are greater than any other term.

Case II : When is not an integer.
Let m be the integral part of , i.e. Then from (I),  Tr + 1 > Tr for r = 1, 2, 3, ....., m and Tr + 1 < Tr for r = m + 1, m + 2, .......
Hence, in this case the Tm + 1 is the greatest term.

### PROPERTIES OF BINOMIAL COEFFICIENTS

We have Also, Let us denote by C0, C1, C2,...., Cn respectively.
Then the above expressions become and C0, C1, C2,....,Cn are called the binomial coefficient and have the following properties:
1. In the expansion of (1 + x)n the coefficient of terms equidistant from the beginning and end are equal.
The coefficient of (r + 1)th term from the beginning is nCr. The (r + 1)th term from the end is (n – r + 1)th term from the beginning. Therefore, its coefficient is nCn – r.
But nCr = nCn – r
Hence the coefficient of terms equidistant from the beginning and end are equal.
1. The sum of the binomial coefficient in the expansion of (1 + x)n is 2n.
Putting x = 1 in (1 + x)n = C0 + C1 x + C2 x2 + ..... + Cr xr + .... + Cn xn, we get
C0 + C1 + C2 + ..... + Cn = 2n or .
1. The sum of the coefficient of the odd terms is equal to the sum of the coefficient of the even terms and each is equal to
2n – 1 i.e., C0 + C2 + C4 + ...... = C1 + C3 + C5 + ..... = 2n – 1
Putting x = 1 and –1 respectively in the expansion. ,
we get C0 + C1 + C2 + C3 + ..... + Cn – 1 + Cn = 2n
and C0 – C1 + C2 – C3 + .... + (–1)n Cn = 0
Adding and subtracting these two equations, we get
C0 + C2 + C4 + .... = C1 + C3 + C5 + .... = 2n – 1
1 . C1 + 2 . C2 + 3 . C3 + .... + n . Cn = n . 2n – 1 or
we have Differentiating both sides w.r.t. x and putting x = 1, we get EXTRA IMPORTANT RESULTS

1. 2. that is, and 3. 4. 5. 6. 7. 8. 9.  ## Want to know more

Want to Know More
Please fill in the details below:

## Latest NEET Articles\$type=three\$c=3\$author=hide\$comment=hide\$rm=hide\$date=hide\$snippet=hide

Name

Alternating Current,60,Basic Maths,2,best books for iit jee,2,best coaching institute for iit,1,best coaching institute for iit jee preparation,1,best iit jee coaching delhi,1,best iit jee coaching in delhi,2,best study material for iit jee,4,Blog,62,books for jee preparation,1,books recommended by iit toppers,3,Capacitance,3,CBSE NEET,9,cbse neet 2019,3,CBSE NEET 2020,1,cbse neet nic,1,Centre of Mass,2,Chemistry,58,Class 12 Physics,15,coaching for jee advanced,1,coaching institute for iit jee,2,Collision,2,Current Electricity,4,Electromagnetic Induction,3,Electronics,1,Electrostatics,3,Energy,1,Fluid Mechanics,4,Gravitation,2,Heat,4,iit admission,1,iit advanced,1,iit coaching centre,3,iit coaching centre in delhi,2,iit coaching classes,2,iit coaching in delhi,1,iit coaching institute in delhi,1,iit entrance exam,1,iit entrance exam syllabus,2,iit exam pattern,2,iit jee,4,iit jee 2019,3,iit jee advanced,2,iit jee books,3,iit jee coaching,2,iit jee exam,3,iit jee exam 2019,1,iit jee exam pattern,3,iit jee institute,1,iit jee main 2019,2,iit jee mains,2,iit jee mains syllabus,2,iit jee material,1,iit jee online test,3,iit jee practice test,3,iit jee preparation,6,iit jee preparation in delhi,2,iit jee preparation time,1,iit jee preparation tips by toppers,2,iit jee question paper,1,iit jee study material,3,iit jee study materials,2,iit jee syllabus,2,iit jee syllabus 2019,2,iit jee test,3,iit preparation,2,iit preparation books,5,iit preparation time table,2,iit preparation tips,2,iit syllabus,2,iit test series,3,IITJEE,100,JEE Advanced,82,jee advanced exam,1,jee advanced exam pattern,1,jee advanced paper,1,JEE Books,1,JEE Coaching Delhi,3,jee exam,3,jee exam 2019,6,JEE Exam Pattern,2,jee exam pattern 2019,1,jee exam preparation,1,JEE Main,85,jee main 2019,4,JEE Main application form,1,jee main coaching,1,JEE Main eligibility criteria,3,jee main exam,1,jee main exam 2019,3,jee main online question paper,1,jee main online test,3,jee main registration,2,jee main syllabus,2,jee mains question bank,1,jee mains test papers,3,JEE Mock Test,2,jee notes,1,jee past papers,1,JEE Preparation,2,jee preparation in delhi,1,jee preparation material,4,JEE Study Material,1,jee syllabus,6,JEE Syllabus Chemistry,1,JEE Syllabus Maths,1,JEE Syllabus Physics,1,jee test series,3,Kinematics,1,Latest article,1,Latest Articles,61,Latest News,1,latest news about neet exam,1,Laws of Motion,2,Magnetic Effect of Current,3,Magnetism,3,Modern Physics,1,NCERT Solutions,15,neet,2,neet 2019,1,neet 2019 eligibility criteria,1,neet 2019 exam date,2,neet 2019 test series,2,NEET 2020 Eligibility Criteria,1,neet application form 2019 last date,1,Neet Biology Syllabus,1,Neet Books,3,neet eligibility criteria,3,neet exam 2019,7,neet exam application,1,neet exam date,1,neet exam details,1,neet exam pattern,6,neet exam pattern 2019,2,neet examination,1,neet mock test 2019,1,Neet Notes,3,Neet Online Application Form,3,neet online test,2,neet past papers,1,neet physics syllabus,1,neet practice test,2,NEET preparation books,1,neet qualification marks,1,NEET question paper 2019,1,neet question papers,1,neet registration,1,Neet Study Material,3,neet syllabus,6,neet syllabus 2019,5,NEET Syllabus 2020,1,neet syllabus chemistry,1,neet syllabus for biology,1,neet syllabus for physics,1,neet test series,1,neet ug 2019,2,news,2,online study material for iit jee,1,Optical Instruments,1,Physics,110,physics books for iit jee,1,Power,1,Practical Physics,1,Ray Optics,1,Rotational Motion,3,SHM,3,Simple Harmonic Motion,3,study materials for iit jee,1,Study Notes,110,study notes for iit jee,1,Thermodynamics,4,Units and Dimensions,1,Vectors,2,Wave Motion,3,Wave Optics,1,Work,1,
ltr
item
Best IIT JEE Coaching Institute in Delhi | NEET Coaching in East Delhi | NEET Coaching in West Delhi: Binomial Theorem | Mathematics Notes for IITJEE Main
Binomial Theorem | Mathematics Notes for IITJEE Main