Each of the 3 persons is to be given some identical items such that product of the numbers of items received by each of the three persons is equal to 30. In how many maximum different ways can this distribution be done?

We need ordered triplets (x, y, z) of positive integers where x × y × z = 30. Prime factorization of 30 = 2 × 3 × 5. All unordered factor sets and their arrangements: (2, 3, 5) — all distinct → 3! = 6 ways (1, 5, 6) — all distinct → 3! = 6 ways (1, 3, 10) — all distinct → 3! = 6 ways (1, 2, 15) — all distinct → 3! = 6 ways (1, 1, 30) — two same → 3!/2! = 3 ways Total: 6 + 6 + 6 + 6 + 3 = 27 ways.