Binomial theorem

Binomial coefficients give the expansion of an integer nonnegative power of a sum.

(a+b)^n=\sum_{k=0}^n\binom nk a^{n-k}b^k

Different ways to understand it

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Combinations
Choose k distinct elements from an n-element set without order, for integers $0\le k\le n$ . Each chosen subset corresponds to k!( $n-k$ )! full permutations. Repetition-allowed selections are a different counting problem and need their own assumptions.
Inclusion–exclusion
Alternating intersection counts correct double-counting in unions of finite sets.
Recurrence relations
A recurrence specifies sequence terms using earlier terms and initial data.
Burnside's lemma
For a finite group action, orbit count equals the average number of fixed points.
Catalan numbers
Catalan numbers count several recursively structured objects, including balanced parenthesis strings.