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**If x, y ∈ R and n ∈ N, then the binomial theorem states that**

(x+y)^{n}=^{n}C_{0}x^{n}+^{ n}C_{1}x^{n-1}y+^{n}C_{2}x^{n-2}y^{2 }+…… … .. +^{n}C_{r}x^{n-r}y^{r }+ ….. +^{n}C_{n}y^{n}

which can be written as Σ^{n}C_{r}x^{n-r}y^{r}. This is also called as the binomial theorem formula which is used for solving many problems.**Some chief properties of binomial expansion of the term (x+y)**^{n}:

1. The number of terms in the expansion is (n+1) i.e. it is one more than the index.

2. The sum of indices of x and y is always n.

3. The binomial coefficients of the terms which are equidistant from the starting and the end are always equal. The simple reason behind this is

C(n, r) = C(n, n-r) which gives C(n, n) C(n, 1) = C(n, n-1) C(n, 2) = C(n, n-2).**Such an expansion always follows a simple rule which is:**

1. The subscript of C i.e. the lower suffix of C is always equal to the index of y.

2. Index of x = n – (lower suffix of C).- The (r +1)
^{th}term in the expansion of expression (x+y)^{n}is called the general term and is given by T_{r+1}=^{n}C_{r}x^{n-r}y^{r} - The term independent of x is obviously without x and is that value of r for which the exponent of x is zero.
- The middle term of the binomial coefficient depends on the value of n. There can be two different cases according to whether n is even or n is odd.

1. If n is even, then the total number of terms are odd and in that case there is a single middle term which is (n/2 +1)^{th}and is given by^{n}C_{n/2 }a^{n/2 }x_{n/2}.

2. On the other hand, if n is odd, the total number of terms is even and then there are two middle terms [(n+1)/2]^{th }and [(n+3)/2]^{th}which are equal to^{ n}C_{(n-1)/2 }a^{(n+1)/2}x^{(n-1)/2}and^{n}C_{(n+1)/2}a^{(n-1)/2}x^{(n+1)/2} - The binomial coefficient of the middle term is the greatest binomial coefficient of the expansion.
- Some of the standard binomial theorem formulas which should be memorized are listed below:

1. C_{0 }+ C_{1 }+ C_{2}+ ….. + C_{n}= 2^{n}

2. C_{0 }+ C_{2}+ C_{4}+ ….. = C_{1}+ C_{3}+ C_{5}+ ……….= 2^{n-1}

3. C_{0}^{2}+ C_{1}^{2 }+ C_{2}^{2}+ ….. + C_{n}^{2}= 2_{n}C_{n }= (2n!)/ n!n!

4. C_{0}C_{r}+ C_{1}C_{r+1 }+ C_{2}C_{r+2}+ ….. + C_{n-r}C_{n}=(2n!)/ (n+r)!(n-r)!

5. Another result that is applied in binomial theorem problems is^{n}C_{r }+^{n}C_{r-1}=^{n+1}C_{r}

6. We can also replace^{m}C_{0}by^{m+1}C_{0}because numerical value of both is same i.e. 1. Similarly we can replace mCm by^{m+1}C_{m+1}.

7. Note that (2n!) = 2^{n}. n! [1.3.5. … (2n-1)] - In order to compute numerically greatest term in a binomial expansion of (1+x)
^{n}, find T_{r+1}/ T_{r}= (n – r + 1)x/r. Then put the absolute value of x and find the value of r which is consistent with the inequality T_{r+1}/ T_{r}> 1. - If the index n is other than a positive integer such as a negative integer or fraction, then the number of terms in the expansion of (1+x)
^{n }is infinite. - The expansions in ascending powers of x are valid only if x is small. If x is large, i.e. |x| > 1 then it is convenient to expand in powers of 1/x which is then small.
- The binomial expansion for the nth degree polynomial is given by:
- Following expansions should be remembered for |x| < 1:

1. (1+x)^{-1}= 1 – x + x^{2 }– x^{3}+ x^{4}^{-}….. ∞

2. (1-x)^{-1 }= 1 + x + x^{2}+ x^{3 }+ x^{4}+ ….. ∞

3. (1+x)^{-2}= 1 – 2x + 3x^{2 }– 4x^{3}+ 5x^{4}- ….. ∞

4. (1-x)^{-2}= 1 +2x + 3x^{2}+4x^{3 }+ 5x^{4}+ ….. ∞

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