Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all the five balls. In how many ways can we place the balls so that no box remains empty?

Options

A. `""^(5) C_3`
B. `5!`
C. `150`
D. `5^3`

Answer

To solve the problem of placing five balls of different colors into three boxes of different sizes such that no box remains empty, we can use the principle of inclusion-exclusion. Here's a step-by-step solution:

### Step 1: Calculate the total arrangements without restrictions
First, we calculate the total number of ways to place the 5 balls into 3 boxes without any restrictions. Since each ball can go into any of the 3 boxes, the total arrangements are given by:
\[ 3^5 = 243 \]
This is because each ball has 3 choices (one for each box).

**Hint:** Remember that when you have multiple independent choices, you multiply the number of options for each choice.

### Step 2: Subtract arrangements where at least one box is empty
Next, we need to subtract the cases where at least one box is empty. We can use the principle of inclusion-exclusion for this.

#### Step 2.1: Calculate the cases where at least one box is empty
1. **Choose 1 box to be empty:** There are 3 ways to choose which box is empty. The remaining 5 balls can then go into the 2 remaining boxes. The number of arrangements in this case is:
\[ 2^5 = 32 \]
So, the total for this case is:
\[ 3 \times 32 = 96 \]

**Hint:** When one box is empty, you only have the remaining boxes to place all the balls.

#### Step 2.2: Add back arrangements where at least two boxes are empty
2. **Choose 2 boxes to be empty:** There are 3 ways to choose which 2 boxes are empty. All 5 balls must go into the remaining box, which can only be done in 1 way. So, the total for this case is:
\[ 3 \times 1 = 3 \]

**Hint:** If only one box is left, all balls must go into that box, which gives you only one arrangement.

### Step 3: Apply inclusion-exclusion principle
Now, we can apply the inclusion-exclusion principle:
\[ \text{Total arrangements with no box empty} = \text{Total arrangements} - \text{At least one box empty} + \text{At least two boxes empty} \]
Substituting the values we calculated:
\[ \text{Total arrangements with no box empty} = 243 - 96 + 3 = 150 \]

### Final Answer
Thus, the total number of ways to place the five balls in the three boxes such that no box remains empty is:
\[ \boxed{150} \]