**Binomial Theorem:**

General form of a polynomial in x is \(a_n x^n~+~a_{n-1} x^{n-1}~ + ⋯..+~a_1 x~+~a_0\), where \(a_n, a_{n-1}, …..,a_1, a_0\) are constants, \(a_n\) ≠ 0 and n is a whole number. A binomial is a polynomial which consists of two terms. Algebraically, sum or difference of two monomials is a binomial. A binomial expression is given by \(ax^n~+~bx^m\) , where a,b are non- zero real coefficients and m,n are distinct non- negative integers.

**Example: \( 5x^2~-~3x,x^4~-3~x^3~+~2x~+~8\)**

We all know the expansion of algebraic identities such as: \((a+b)^2, (a+b)^3, (a-b)^2, (a-b)^3\).

But if the exponent raised to a binomial expression is large comparatively say 50, then expansion by identities becomes too cumbersome. In such cases, binomial theorem or binomial expansion is used which gives the algebraic expansion of a binomial raised to any exponent.

According to binomial theorem, expansion of \((a+b)^n\) is given as:

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

where, \( {n \choose k}~ =~ nC_k~ =~C(n,k)~=~\frac{n!}{(n-k)!k!}\) is the binomial coefficient in the above theorem. It represents the number of ways in which k unordered outcomes can be selected from a total of n possibilities.

The above formula can be expanded as following:

\((a+b)^n~=~{n \choose 0} a^n b^0~+~{n \choose 1} a^{n-1} b^1~+~{n \choose 2} a^{n-2} b^2~+~………{n \choose n-1} a^1 b^{n-1}~+~{n \choose n} a^0 b^n \)

Let us write the expansion of \((a+b)^n\), [ 0 ≤ n ≤ 5 and n is an integer] and have a look at the properties of the binomial expansion.

\((a+b)^0\)=\(1\)

\((a+b)^1\)=\((a+b)\)

\((a+b)^2\)=\(a^2+2ab+b^2\)

\((a+b)^3\)=\(a^3+3a^2 b+3ab^2+b^3\)

\((a+b)^4\)=\(a^4+4a^3 b+6a^2 b^2+4ab^3+b^4\)

\((a+b)^5\)=\(a^5+5a^4 b+10a^3 b^2+10a^2 b^3+5a^4 b+b^5\)

**Properties of the expansion:**

- The exponent of a is decreasing by 1 in the successive terms and exponent of b is increasing by 1 in the successive terms.

For example, in the expansion of

\( (a+b)^4~=~a^4~+~4a^3b~+~6a^2b^2~+~4ab^3~+~b^4\)

It can be observed that the exponent of a decreases with every successive term and the exponent of b increases successively and also observe the nature of binomial coefficients as C(n,k) = C(n,n – k). As it can be seen that the coefficients of first and last term are same, also second and fourth terms have the same coefficients.

- In the expansion of \( (a~+~b)^n \), total number of terms is one more than exponent of binomial expression, i.e., if exponent of a binomial is , then total number of terms in the expansion is n+1.

For example, in the expansion of

\( (a+b)^5~=~a^5~+~5a^4b~+~10a^3b^2~+~10a^2b^3~+~5a^4b~+~b^5\)

Exponent of (a + b) is 5 and the number of terms in the expansion is 6.

- In each term of the expansion of \( (a + b)^n\) , the sum of exponents will always be equal to n .

Let us now look into relation between exponent of (a + b) and coefficients of the terms in expansion of \( (a + b)^5\) .

- The coefficients of the first term and last term of the expansion \( (a~+~b)^n\)[where n can be any integer] is 1.
- The binomial coefficient of each exponent is the sum of entries above it . As it can be seen the coefficient of index 2 is given by sum of coefficients of 1 lying above it i.e., 1 + 1 and so on. This can be visualized easily by the representation given below:

Let us see what happens if we try to add the coefficients of the expansion.

If we take the fifth row i.e.n = 5 , it can be seen that there are six terms and if we add the coefficients then sum, s = 32 which is equal to \(2^5\).

Similarly on taking the fourth row i.e. n = 4 , the sum comes out to be 16 which is again equal to \( 2^4\)

For third row also the sum is 8 which is equal to \( 2^3 \) .

Thus it can be said that for expansion of \( (a~+~b)^n \) , the sum of binomial coefficients is equal to \( 2^n \).

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