A binomial is an algebraic expression with two dissimilar terms connected by + or – sign. Let’s look into the following example to understand the difference between monomial, binomial and trinomial.
Note: Binomial Theorem is different than binomial distribution.
Binomial Theorem is a quick way of expanding a binomial expression with (that are raised to) large powers. This theorem is a really important topic(section) in algebra and has application in Permutations and Combinations, Probability, Matrices, and Mathematical Induction. If you are preparing for competitive exams for university admission or for jobs then this theorem is really important for you as it is a basic and important section of algebra. In this chapter, you will learn a shortcut that will allow you to find (x + y)^{n} without multiplying the binomial by itself n times.
Special cases of the binomial theorem were known since at least the 4th century BC when Greek Mathematician Euclid mentioned the special case of the binomial theorem for exponent 2. The binomial theorem for cubes was known by the 6th century in India. Isaac Newton is generally credited with the generalized binomial theorem, valid for any rational exponent.
First, look into some following identities that you have done earlier
Can you guess the next expansion for the binomial (x + y)^{5}?
Now, you will observe that from above:
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.
The coefficients of the terms in the extension are the binomial coefficients.
If we arrange the coefficients in these expansions, this will look like
The structure given in the above figure looks like a triangle with 1 at the top vertex and running down the two slanting sides. The numbers in this array are known as Pascal’s triangle, after the name of French mathematician Blaise Pascal. It is also known as Meru Prastara by Pingla.
Expanding a binomial with a high exponent such as (x + 2y)^{16} can be a lengthy process. Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term. Note the pattern of coefficients in the expansion of (x + y)^{5}
The second term is . The third term is . In this way, we can generalize this result.
The (r + 1)th term of the binomial expansion of (x + y)^{n} is: .
The middle term in the expansion (a + b)^{n} , depends on the value of 'n'.
When 'n' is even
In this case, the number of terms in the expansion will be n + 1. Since n is, even so, n + 1 is odd. Therefore, the middle term is term.
It is given by .
When 'n' is odd
In this case, the number of terms in the expansion will be n + 1. Since n is, odd so, n + 1 is even. Therefore, there will be two middle terms in the expansion, namely and term.
And it is given by and .
e = 2.71828182846.......(digit goes on without repeating)
Let's use Binomial Theorem:
now for a most accurate value of e, n should be as big as possible ( is representing that n should be the biggest possible number)
Now,
So the equation remains,
With calculating just a few terms we get e = 2.7083
The last expression is calculated using supercomputers to get the most accurate value of e.
we know that 32 = 2^{5}, so, 32^{30} can be written as
(2^{5})^{30 } = 2^{150} = (2^{3})^{50} = 8^{50} = (7 + 1)^{50}
= [ (7)^{50} + ^{50}C_{1}(7)^{49} + ^{50}C_{2} (7)^{48} + ... + 1 ]
= [ 7( (7)^{49} + ^{50}C_{1}(7)^{48} + ^{50}C_{2} (7)^{47} + ... ) + 1 ]
= 7k + 1
⇒ remainder is 1.
The above theorem can be used in solving problems such as, which one is greatest among 100^{100} and (300)! ?
From the above results,
Put n = 300
= (100)^{300} < (300)! .........(i)
But, = (100)^{300} > (100)^{100} .........(ii)
From (i) and (ii)
= (100)^{100} < (100)^{300} < (300)!
Hence, (100)^{100} < (300)!
First, you need to understand Binomial Theorem. You should be able to find the general term, greatest term, of any binomial equation. After getting a strong understanding of the theorem, you can jump on the application part.
If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.
Start from NCERT book, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.
Then you can refer to the book Cengage Mathematics Algebra. The binomial theorem is explained very well in this book and there are lots of questions with crystal clear concepts. You can also refer to the book Arihant Algebra by SK Goyal or RD Sharma. But again the choice of reference book depends on person to person, find the book that best suits you the best depending on how well you are clear with the concepts and the difficulty of the questions you require.
Chapters 
Chapters Name 
Chapter 1 

Chapter 2 

Chapter 3 

Chapter 4 

Chapter 6 

Chapter 7 

Chapter 8 

Chapter 9 

Chapter 10 

Chapter 11 

Chapter 12 

Chapter 13 

Chapter 14 

Chapter 15 

Chapter 16 
The coefficient of x^{−5} in the binomial expansion of
,
where , is :
1
4
4
1
The value of
2^{21}−2^{10 }
2^{20}−2^{9 }
2^{20}−2^{10 }
2^{21}−2^{11}
If the number of terms in the expansion of
28, then the sum of the coefficients of all the terms in this expansion, is
64
2187
243
729