Let S = 1, 2, 3, , 10 . Consider the set X = R : R is an equivalence relation on the set S such that R has exactly 42 elements . Then the number of elements in X is _____________.

Step-by-step solution

An equivalence relation on a set S corresponds to a partition of S into disjoint equivalence classes. Let the sizes of these equivalence classes be a₁, a₂, , a_k . The number of elements in the equivalence relation R is given by the sum of the squares of the sizes of its equivalence classes: _ i=1 ^k a_i^2 = 42 Since the total number of elements in S is 10 , we also have: _ i=1 ^k a_i = 10 We need to find all possible partitions of 10 such that the sum of their squares is 42 . Let's check the possible sizes of the