Binomial Theorem Proof | Derivation of Binomial Theorem Formula

What is the Binomial Theorem?

Binomial Theorem is a quick way of expanding binomial expression that has been raised to some power generally larger. Generally multiplying an expression – (5x – 4)10 with hands is not possible and highly time-consuming too. Thankfully, Mathematicians have figured out something like Binomial Theorem to get this problem solved out in minutes. You just have to put the values in the binomial expansion formula to find the answer.

The formal expression of the Binomial Theorem is as follows:

\[(a+b)^n~=~∑_{k=0}^n~ {n \choose k} ~a^{n-k} b^k\]

Yeah, I know you must have seen this formula earlier and used too. This is the Binomial Theorem Formula or Binomial expansion formula that means the same thing. However, the notations could be different. The other popular form of the binomial theorem is given as below –

\[ {n \choose k}~ =~ nC_k~ =~C(n,k)~=~\frac{n!}{(n-k)!k!}\]

If you have studied factorial then you must be sure of notation “n!” in mathematics where you had to write all numbers between 1 to n. When putting values in the formula, it would be given as –

\[ ~ 10C_7~ = ~\frac{10!}{(10-7)!7!} = ~\frac{10!}{3!7!}== ~\frac{1*2*3*4*5*6*7*8*9*10}{1*2*3*1*2*3*4*5*6*7}\]

Binomial Theorem

Multiplying binomials together is easy but numbers become more than three then this is a huge headache for the users. Luckily, we have the binomial theorem to solve the large power expression by putting values in the formula and expand it properly. In mathematics, FOIL is a popular technique that is used to multiple two binomial expressions together. When multiplying two binomial equations, the final answer is the Trinomial.

Consider the example of the expression (x + 1)3that can be expanded as x + 1)(x + 1)(x + 1). Now start solving the equation through FOIL method where multiplying two equations will result in x^2 + 2x + 1. Two expressions are multiplied already but (x+1) is still hanging on the head. For this purpose, you need to apply super FOIL technique and add some more steps here.

In brief, the overall process is highly complex and time-consuming as the number of terms to be multiplied together will increase. The FOIL multiplication technique can be continued as long as possible using super duper multiplication technique but it is not favorable for incredibly large powers. The best idea is to use the Binomial Theorem to make the things easier for you.

Binomial Theorem Formula

There is another technique to find the value of “nCr”, and it’s called “Pascal’s Triangle”. For this purpose, you need to design a Pyramid first. Here, is give the pyramid of one as shown below –
Binomial Theorem

Binomial Theorem

Binomial Theorem

\[\large (x+y)^{n} = x^{n} + nx^{n-1}y + \frac{n(n-1)}{2!} x^{n-2} y^{2} + … + y^{n}\]

The next row you will get by adding the pairs of numbers above and put the value in the next row in the middle. Every time you have to expand this way, so Pascal Triangle too is very lengthy process or just use the calculator to make the things manageable.

For some exams, calculators are not allowed especially when you appear for the competitive exams. So, the best idea is to get a depth understanding of the topic by reading the information given in the post and explore your learning with us.


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