Problem:Let n be a positive integer, and let \(S_n = {n^2 + 1, n^2 + 2, ... , (n + 1)^2}\). Find, in terms of n, the cardinality of the set of pairwise products of distinct elements of \(S_n\)

For example, \(S_2 = {5, 6, 7, 8, 9},\)

5 × 6 = 6 × 5 = 30,

5 × 7 = 7 × 5 = 35,

5 × 8 = 8 × 5 = 40,

5 × 9 = 9 × 5 = 45,

6 × 7 = 7 × 6 = 42,

6 × 8 = 8 × 6 = 48,

6 × 9 = 9 × 6 = 54,

7 × 8 = 8 × 7 = 56,

7 × 9 = 9 × 7 = 63,

8 × 9 = 9 × 8 = 72,

and the required cardinality is 10.

I have not been able to solve it.

Apologies for not posting the problem here as could not resolve latex issues. Someone please post the problem here.