The Binomial Theorem
A complete guide: from basic expansions to Newton's generalized series
1. Statement & Notation
The Big Idea
How do you expand $(a+b)^n$ without multiplying the bracket out $n$ times? The Binomial Theorem gives a closed-form answer in one line.
Here the binomial coefficient is defined as
$$\binom{n}{k} = \frac{n!}{k!\,(n-k)!}, \qquad 0\le k\le n.$$It counts the number of ways to choose $k$ objects from $n$ objects (order irrelevant, no repeats). Common alternative notations include $C(n,k)$, ${}^{n}C_{k}$, $C_{k}^{n}$, and $\displaystyle\binom{n}{k}$ (read "$n$ choose $k$").
Writing out the first few powers explicitly:
| $n$ | Expansion of $(a+b)^n$ | Terms |
|---|---|---|
| 0 | $1$ | 1 |
| 1 | $a+b$ | 2 |
| 2 | $a^2+2ab+b^2$ | 3 |
| 3 | $a^3+3a^2b+3ab^2+b^3$ | 4 |
| 4 | $a^4+4a^3b+6a^2b^2+4ab^3+b^4$ | 5 |
2. Pascal's Triangle
Construction
Pascal's triangle is built row by row. Each row starts and ends with $1$, and every interior entry is the sum of the two entries diagonally above it. This rule is precisely Pascal's recurrence:
$$\boxed{\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}}, \qquad 1\le k\le n-1.$$with boundary conditions $\binom{n}{0}=\binom{n}{n}=1$.
How to read the triangle
- Row $n$ contains the coefficients of $(a+b)^n$.
- Entry $k$ in row $n$ is $\binom{n}{k}$.
- Example: Row 4 is $1,4,6,4,1$ — exactly the coefficients of $a^4+4a^3b+6a^2b^2+4ab^3+b^4$.
Historical note
The triangle was known centuries before Pascal — it was discovered in India (Pingala, ~200 BCE) and China (Jia Xian, ~1050 CE). Pascal systematically studied its properties in 1654.