Properties of the binomial coefficient

The binomial coefficient $\dbinom{n}{k}$ satisfies three fundamental identities, all visible in Tartaglia’s triangle.

Property — Identities of the binomial coefficient

\binom{n}{k} &= \binom{n}{n-k} && \text{(simmetria)} \\[4pt] \binom{n}{k} &= \binom{n-1}{k-1} + \binom{n-1}{k} && \text{(formula di Tartaglia/Stifel)} \\[4pt] \sum_{k=0}^{n} \binom{n}{k} &= 2^n && \text{(somma della riga di Tartaglia)} \end{aligned}$$

Symmetry says that choosing $k$ objects to keep is equivalent to choosing the $n-k$ to discard. The Tartaglia/Stifel formula is the rule by which the triangle is built: each number is the sum of the two above it. The third identity states that the sum of a whole row equals $2^n$, that is, the total number of subsets of a set of $n$ elements.

Links

Topics: Combinatorics
Concepts: Binomial coefficient · Tartaglia’s triangle
Methods: Tartaglia triangle
Skills: Combinatorial calculus · Using formulae
People: Stifel · Niccolò Tartaglia