Binomial Theorem
The Binomial Theorem gives a formula for expanding $(a + b)^n$ as a sum of terms involving binomial coefficients.
Formula
(a+b)^n = \sum_{k=0}^{n} C(n,k) a^{n-k} b^k
Definition
The Binomial Theorem is a shortcut for expanding expressions like $(a + b)^n$, using a formula with coefficients from Pascal's Triangle instead of multiplying $(a+b)$ by itself $n$ times. Formally, $(a+b)^n = \sum_{k=0}^{n} C(n,k) a^{n-k} b^k$, where $C(n,k) = n!/(k!(n-k)!)$ is the binomial coefficient, giving the number of ways to choose $k$ items from $n$; the sum of all coefficients is $2^n$ (setting $a = b = 1$). It can be proved by induction or combinatorially (since $C(n,k)$ counts the ways to choose $k$ copies of $b$ from $n$ factors), and Newton's generalized binomial theorem extends it to non-integer and even complex exponents: $(1+x)^\alpha = \sum_{k=0}^{\infty} C(\alpha,k) x^k$ for $|x| < 1$.
Example
$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$, with coefficients $1, 3, 3, 1$ from the $4$th row of Pascal's Triangle. For $(2x - 3)^4$, the $k=2$ term is $C(4,2)(2x)^2(-3)^2 = 6 \times 4x^2 \times 9 = 216x^2$, and the full expansion is $16x^4 - 96x^3 + 216x^2 - 216x + 81$. Newton's binomial series approximates $\sqrt{1.04}$ via $(1+x)^{1/2} = 1 + x/2 - x^2/8 + \ldots$ with $x=0.04$: about $1 + 0.02 - 0.0002 = 1.0198$, close to the actual $1.0198039\ldots$.
Key Insight
Without the Binomial Theorem, expanding $(a + b)^{10}$ would require multiplying $10$ times; with it, you write out $11$ terms instantly, and the middle term (when $n$ is even) has the largest coefficient, reflecting the maximum of the binomial distribution at its mean. Newton's generalized binomial series is a power series convergent for $|x| < 1$, a precursor to Taylor series that reveals how binomial coefficients extend naturally to the complex numbers via the Gamma function.