How many different equivalence relations with exactly three different equivalence classes are there on a set with five elements?

Explanation

Equivalence relations on a set are in bijection with partitions of the set. A 5-element set split into 3 classes has partition types (1 + 1 + 3) or (1 + 2 + 2). For the first type, choose the triple: ⁵C₃ = 10. For the second type, choose the singleton (5 ways) and then pair the remaining 4 elements (3 ways): 5 × 3 = 15. Total = 10 + 15 = 25.