Q. Five balls of different colours are to be placed in three different boxes such that any box contains at least one ball. What is the maximum number of different ways in which this can be done?
Answer: 150
Notes: Since both balls and boxes are distinct, the total ways without restriction is (3^5 = 243). Using inclusion-exclusion, subtract cases with at least one empty box: (C(3,1) times 2^5 - C(3,2) times 1^5 = 3 times 32 - 3 times 1 = 96 - 3 = 93). Thus, the number of ways with no box empty is (243 - 93 = 150). Hence, the maximum number of different ways is 150.