# Binomial Theorem

**The Expansion of (a + b) ^{n}**

If $n$ is any positive integer, then

$(a + b)^n = a^n + {_nC_1}a^{n - 1}b + {_nC_2}a^{n - 2}b^2 + \, \cdots \, + {_nC_m}a^{n - m}b^m + \, \cdots \, + b^n$

Where

${_nC_m}$ = combination of *n* objects taken *m* at a time.

**Some Example of Binomial Expansion**

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

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

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

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

$(a + b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6$

$(a + b)^7 = a^7 + 7a^6b + 21a^5b^2 + 35a^4b^3 + 35a^3b^4 + 21a^2b^5 + 7ab^6 + b^7$

The coefficient of terms can also be found by

**Properties of Binomial Expansion**

- The first term and last term of the expansion are $a^n$ and $b^n$, respectively.
- There are $n + 1$ terms in the expansion.
- The sum of the exponents of $a$ and $b$ in any term is $n$.
- The exponent of $a$ decreases by $1$, from $n$ to $0$.
- The exponent of $b$ increases by $1$, from $0$ to $n$.
- The coefficient of the second term and the second from the last term is $n$.

**Pascal's Triangle**

Pascal's triangle can be used to find the coefficient of binomial expansion.

(a + b)^{0} : 1

(a + b)^{1} : 1 1

(a + b)^{2} : 1 2 1

(a + b)^{3} : 1 3 3 1

(a + b)^{4} : 1 4 6 4 1

(a + b)^{5} : 1 5 10 10 5 1

(a + b)^{6} : 1 6 15 20 15 6 1

(a + b)^{7} : 1 7 21 35 35 21 7 1

**r ^{th} term of (a + b)^{n}**

or

where *m* = *r* - 1

For *n* = even, the middle term is at