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Introduction
A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'.
For n = 1,2,3,4, the expansion of (a + b)n, has been expressed in a very systematical manner in terms of combinatorial coefficients. The above expression suggest the conjecture that (a + b)n should be expressible in the form
for every natural number n.
Principle of Mathematical Induction
If P(n) is a statement (nÎN); such that
1. P(1) is true and 2. truth of P(k) implies the truth of P(k+1), then by the principle of mathematical induction (P.M.I.), the statement P(n) is true for nÎN.
Alternative Proof of Binomial Theorem for Positive Integral Index (Combinatorial Method)
We have, (a + b)n = (a + b) (a + b) ....... n times.
The terms on the RHS are obtained by taking one letter from each factor and multiplying them together.
Choosing 'a' from all the factors, we get the term an.
Method of writing expansion
The first term in the expansion of (a + b)n is nC0anb0. For the second term, the coefficient is taken as nC1, the power of 'a' is decreased by one and the power of 'b' is increased by one. So, the second term is nC1an-1b1. For the third term, the coefficient is taken as nC2, the power of 'a' is again decreased by one and the power of 'b' is increased by one. This process goes on, till we get the last term as nCna0bn.
Some particular expansions for Positive Integral Index
Working rules for expanding (a + b)nnÎN:
Step 1: The value of index, n implies that there will be n+1 terms in the expansion of (a + b)n.
Step 2: Write the first term: nC0anb0.
Step 3: For the second term, take coefficient as nC1, decreases the power of 'a' by 1 and increases the power of 'b' by 1. Thus,
the second term is nC1an-1b1.
Step 4: For the third term, take coefficient as nC2, power of 'a' as n-2 and power of 'b' as 2. Continue this process repeatedly till the last term nCna0bn is obtained.
Step 5: For evaluating nCr, it is useful to write nCr as nCn-r, if r > n/2.
General Term for Positive Integral Index
General Term for Positive Integral Index is:
For 0 
r 
n, we have Tr+1=nCran-rbr.
Middle Terms for Positive Integral Index
The number of terms in the expansion of (a + b)n depends on the index n. The index n is either even or odd.
Particular Terms for Positive Integral Index
Sometimes, a particular term satisfying certain conditions is required in the binomial expansion of the type (a + b)n. This can be done by expanding (a + b)n and then locating the required term. Generally this becomes a tedious task, specially when the index n is large. In such cases, we begin by evaluating the general term Tr+1 and then finding the value of r by assuming Tr+1 to be the required term
Greatest Terms for Positive Integral Index
Working rules for finding the greatest term:
Step 1: In (a + b)n, the constants a and b must be positive.
Step 2: Write Tr+1 and Tr and find the value of Tr+1/Tr.
Step 3:Simplify the inequality (Tr+1/Tr) greater than or equal to 1 and find the greatest possible value of r satisfy this inequality
Step 4: Calculate Tr+1 for this value of r. This gives the greatest term.
Some Applications of Binomial Theorem for Positive Integral Index
nC0, nC1, ..... nCn are called binomial coefficients. nC0, nC2 nC4, ..... are called even binomial coefficients. nC1, nC3, nC5 .... are called odd binomial coefficients.
In case of no ambiguity, the binomial coefficients nC0, nC1, ..... nCn are written as C0, C1, ..... Cn.
Binomial Theorem for Fractional Index
For any rational number n,
We accept this expansion without proof.
Some particular expansions for Fractional Index
For n 
Q, we have:
General Term for Fractional Index
The General Term for Fractional Index is: 
Particular Terms for Fractional Index
Sometimes, a particular term satisfying certain conditions is required in the binomial expansion of the type (1+x)n. This can be done by expanding (1+x)n to certain terms and then locating the required term. Generally this becomes a tedious task. In such cases, we begin by evaluating the general term Tr+1 and then finding the value of r by assuming Tr+1 to be the required term.
Some Applications of Binomial Theorem for Fractional Index
If x be numerically so small that its square and higher powers may be neglected, then (1+x)n= 1+nx (approximately), because x2, x3, x4,…. are all approximately zero.
Summary
1. A sentence is called a statement if it can be adjudged as true or false. Every statement is a sentence, but a sentence may or may not be a statement.
2. A statement involving natural number n is generally denoted by P(n).
3. A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'.
