Binomial Theorem

The Binomial Theorem is a special tool for expanading expressions of the form:

\[(\text{something} + \text{somethingelse})^{\text{fixed\_number}} \\ (a + b)^n\]

The theorem states:

\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \\ =\binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1}b^1 + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n}a^0b^n\]

where:

• \(n\) is a fixed positive integer.
• The term \(\binom{n}{k}\) is the binomial coefficient, calculated as: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)