1. The problem: Understand the Binomial Theorem in a simple way. 2. The Binomial Theorem helps us expand expressions like $$(a+b)^n$$ without multiplying the whole thing repeatedly. 3. The formula is: $$ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k $$ where $\binom{n}{k}$ is the binomial coefficient, meaning "n choose k," which counts how many ways to pick $k$ items from $n$. 4. Important rules: - $n$ is a non-negative integer. - $a$ and $b$ are any numbers or expressions. - The binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ where $!$ means factorial. 5. Step-by-step example: Expand $$(a+b)^3$$ - Calculate coefficients: $\binom{3}{0} = 1$, $\binom{3}{1} = 3$, $\binom{3}{2} = 3$, $\binom{3}{3} = 1$ - Write terms: $$a^3 b^0, a^2 b^1, a^1 b^2, a^0 b^3$$ - Multiply coefficients and terms: $$1 \cdot a^3 + 3 \cdot a^2 b + 3 \cdot a b^2 + 1 \cdot b^3$$ - So, $$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$ 6. This method saves time and helps find any power expansion quickly. Final answer: The Binomial Theorem formula and how to use it to expand powers of sums.