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