A binomial has the form $(a+b)^n$. When $n$ is a positive integer, it can be expanded according to this formula:

$(a+b)^n = \dbinom{n}{0} a^n b^0 + \dbinom{n}{1} a^{n-1} b^1 + \dbinom{n}{2} a^{n-2} b^2 + ... + \dbinom{n}{n-1} a^1 b^{n-1} + \dbinom{n}{n} a^n b^0$

Pascal's Triangle

The coefficients of the binomial expansion form a pattern known as Pascal's Triangle. (The first row is the 0th row.)



For example, the 4th row of the triangle shows the coefficients for the expansion of $(a+b)^4$.

For high powers of $n$, it is quicker to use the $^nC_r$ method for finding the coefficients, because it takes a while to write out Pascal's Triangle.

Uses of the binomial expansion

The binomial expansion can be used for approximations and calculating binomial probabilities.