A combinatorial proof establishes an identity by showing that both sides count the same finite set.

Suppose a collection of objects can be counted in two different ways. If the first argument gives $A$ objects and the second gives $B$, then

$$A=B$$

because both expressions describe the same cardinality.

Bijection arguments

Another method constructs a bijection between two finite sets. Pairing every object on one side with exactly one object on the other proves that the sets have equal size.

Double counting

For example, the symmetry

$$\binom nk=\binom n{n-k}$$

can be proved by observing that choosing $k$ selected objects is equivalent to choosing the $n-k$ objects left out.

Combinatorial proofs often explain why an algebraic identity is true rather than merely verifying it by symbolic manipulation.