**Binomial Expression**

An expression consisting of two terms, connected by + or – sign is called binomial expression.

**Binomial Theorem**

If a and b are real numbers and n is a positive integer, then

The general term of (r + 1)^{th} term in the expression is given by

T_{r+1} = ^{n}C_{r} a^{n-r }b^{r}

**Some Important Observations from the Binomial Theorem**

The total number of terms in the binomial expansion of (a + b)^{n} is n + 1.

The sum of the indices of a and b in each term is n.

The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficient and

^{n}C_{r} = ^{n}C_{n-r}, r = 0, 1, 2, 3,…, n

The values of the binomial coefficient steadily increase to a maximum and then steadily decrease.

The coefficient of x^{r} in the expansion of (1 + x)^{n} is ^{n}C_{r}.

In the binomial expansion (a + b)^{n}, the r^{th} term from the end is (n – r + 2)^{th} term from the beginning.

**Middle Term in the Expansion of (a + b) ^{n}**

If n is even, then in the expansion of (a + b)

^{n}, the middle term is ( + 1) th term.

If n is odd, then in the expansion of (a + b)^{n}, the middle terms are ()th term and ()th term.