Symmetry (Binomial Coefficient)

combinatorics

Definition

Symmetry of Binomial Coefficients

For , the binomial coefficients satisfy the symmetry identity

Thus choosing elements from an -element set gives the same number of choices as choosing the elements to leave out.

Combinatorial Derivation

The complement map sends every -element subset to the -element subset . Taking the complement again recovers , so this map is bijective. The two classes of subsets therefore have equal size.

Example

Example