Find the total number of binary operations on a set containing 5 elements.

Answer 1

A binary operation on a set S is a map from S x S -> S. If S has 5 elements, then S x S has 25 elements. To construct a map, each of those 25 elements must get mapped to one of the 5 elements in S. So you have to choose 1 from among 5 choices, 25 times. This makes 5²⁵ possibilities.