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?

To find the number of ways to distribute 5 distinct balls into 3 distinct boxes such that each box has at least one ball, we follow these steps: 1. Identify the possible distributions of balls: Since every box must have at least one ball, the two possible distributions are: Case 1: 3 balls in one box, 1 in the second, and 1 in the third (3, 1, 1). Case 2: 2 balls in one box, 2 in the second, and 1 in the third (2, 2, 1). 2. Calculate ways for Case 1 (3, 1, 1): First, choose 3 balls out of 5 for the first box: 5C3 = 10 ways. Then, choose 1 ball out of the remaining 2 for the second box: 2C1 = 2 ways. The last ball goes to the last box: 1C1 = 1 way. Multiply these: 10 x 2 x 1 = 20 ways. Since the 3 balls can be in any of the 3 boxes, multiply by 3: 20 x 3 = 60 ways. 3. Calculate ways for Case 2 (2, 2, 1): First, choose 2 balls out of 5 for the first box: 5C2 = 10 ways. Then, choose 2 balls out of the remaining 3 for the second box: 3C2 = 3 ways. The last ball goes to the last box: 1C1 = 1 way. Multiply these: 10 x 3 x 1 = 30 ways. Since the single ball can be in any of the 3 boxes, multiply by 3: 30 x 3 = 90 ways. 4. Total ways: Add the results from both cases: 60 + 90 = 150 ways. Thus, the correct option is C.