Newton's binomial theorem

The binomial coefficient, born to count, reappears in algebra as the coefficient in the expansion of a power of a binomial.

Theorem — Newton's binomial

For every $a,b\in\mathbb{R}$ and $n\in\mathbb{N}$: $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k}\, a^{n-k} b^k.$ The coefficients $\dbinom{n}{k}$ are exactly the numbers of Tartaglia’s triangle.

Example

$(a+b)^4 = a^4 + 4a^3 b + 6a^2 b^2 + 4ab^3 + b^4$, with coefficients $1,4,6,4,1$ (fifth row of Tartaglia’s triangle).

Links

Topics: Combinatorics
Concepts: Newton’s binomial · Binomial coefficient · Tartaglia’s triangle
Methods: Newton binomial
Skills: Combinatorial calculus · Using formulae
People: Isaac Newton