Use Coupon: CART20 and get 20% off on all online Study Material

Total Price: R

There are no items in this cart.
Continue Shopping
Get instant 20% OFF on Online Material.
coupon code: MOB20 | View Course list

Get extra R 550 off

Introduction To Binomial Theorem

A  Binomial Expression

Any algebraic expression consisting of only two terms is known as a binomial expression. It's expansion in power of x is shown as the binomial expansion.

For example: (i)    a + x             (ii)   a2 + 1/x2       (iii)  4x - 6y

Binomial Theorem

Such formula by which any power of a binomial expression can be expanded in the form of a series is known as binomial theorem.

It can be easily understood by examples.

        (a + x)2 = a2 + 2ax + x2

        (a + x)2 = a3 + 3a2x + 3ax2 + x3

Here, we see that the expression of (a + x)2 is simple, we just multiply (a + x) by (a + x). Expansion of (a + x)3 is little tougher, but what happens when the expansion is raised to the power of ten or more? So, we have to establish the formula for (a + x)n, where n is any integer. Let us define 'a' as the first term, 'x' as the second term and 'n' as the exponent.

The total terms in the expansion of (a + x)2 and (a + x)3 are 3 and 4 respectively, which means that the number of terms in the expansion is one more than the exponent. So total number of terms in the expansion (a + x)n is (n + 1).

        Now, for n = 2

           (a + x)2 = a2x0 +  2a1x1/1 + (2(2-1)/1*2)a0x2

        = (F.T.)n (S.T.)0 + n/1! (F.T.)(n-1) (S.T.)1 + (n(n-1)/2!) (F.T.)(n-2) (S.T.)2

        = a2 + 2ax + x2

Note: F.T. refers to first term i.e. 'a' and S.T. refers to second term i.e. 'x' 


        (a + x)3 = a3 + 3a2x + ((3(3-1)/(1*2))a0 x2 + ((3(3-1)(3-2))/(1*2*3))x3

                     = a3 + 3a2x + 3ax2 + x3

When n is a positive integer, then

(a + x)n = nC0an + nC1an-1x + nC2an-2 x2 +...+ nCran-r xr +...+ nCnxn,

Where nC0 . nC1 . nC2 ... nCn are called Binomial coefficients.

Proof of Binomial Theorem

Proof of Binomial Theorem is very simple; we can prove it by using the mathematical induction.


Step I:      Let n = 1

                L.H.S. = a + x

                R.H.S. = a + 1C1 x = a + x

        So, theorem is true for n = 1.

Step II:

        Let the theorem be true for n = m, than

        P(m) : (a + x)m = mC0 am + mC1 am-1 x1 + mC2 am-2 +...+ mCm xm       (i)

Step III:

        We have to prove for n = m + 1 i.e. we have to prove that

P(m + 1): (a + x)m+1 = m+1C0 am+1 + m+1C1 am x1

+...+ m+1Cm a xm + m+1Cm+1 xm+1             ....(ii)

        Multiplying by (a + x) on both sides in equation (i), we get,

        (a + x)m+1 = (mC0 am+1 + mC1 am +...+ mCm-1 a2 xm-1 + mCm a xm)

                        = (mC0 am x + mC1 mC0) am x +...+ mCm-1 a xm + mCm xm+1)

Or, (a+x)m+1 = mC0 am+1 + (mC1 + mC0) am x +...+ (mCm-1 + mCm-2) a2 xm-1

+ (mCm + mCm-1) a xm + mCm xm+1

                        = m+1C0 am+1 + m+1C1 am x + m+1C2 am-1 x2 +...+

m+1Cm-1 a2 xm-1 + m+1Cm a xm + m+1Cm+1 xm+1.      

Hence, Proved.


        Expand  (x + 1/x)7 .


         (x + 1/x)7 = 7C0x7 + 7C1x6  (1/x) + 7C2x5 (1/x2) + 7C3x4 + 1/x3 + 7C4x4

(1/x3)  + 7C5x2 + (1/x5) + 7C6x (1/x6) + 7C7 (1/x7)

        = x7 + 7x5 + 21x3 + (35 x) + (35/x )+( 21/x3) + (7/x5)+ (1/x7) .

IIT JEE study material is available online free of cost at Study Physics, Chemistry and Mathematics at askIITians website and be a winner. We offer numerous live online courses as well for live online IIT JEE preparation - you do not need to travel anywhere any longer - just sit at your home and study for IIT JEE live online with

To read more, Buy study materials of Binomial Theorem comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.

  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: R 15,000
  • View Details
Get extra R 3,750 off