1. **Problem Statement:** We want to understand the formula of Newton's Binomial Theorem and how it is used to expand expressions of the form $$(a+b)^n$$ where $n$ is a non-negative integer. 2. **Formula:** Newton's Binomial Theorem states: $$ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k $$ where $\binom{n}{k}$ is the binomial coefficient, calculated as: $$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$ 3. **Explanation:** - The binomial coefficient $\binom{n}{k}$ counts the number of ways to choose $k$ elements from $n$. - The term $a^{n-k} b^k$ represents the powers of $a$ and $b$ in each term of the expansion. - The sum runs from $k=0$ to $k=n$, covering all terms. 4. **Important Rules:** - Factorials: $n! = n \times (n-1) \times \cdots \times 1$ with $0! = 1$. - The powers of $a$ decrease from $n$ to $0$ while powers of $b$ increase from $0$ to $n$. 5. **Example:** For $n=2$, $$ (a+b)^2 = \binom{2}{0}a^2b^0 + \binom{2}{1}a^1b^1 + \binom{2}{2}a^0b^2 = a^2 + 2ab + b^2 $$ This formula helps expand binomials without multiplying repeatedly.