Key Concepts and Formulas

• Combinations ($^nC_r$): The number of ways to choose $r$ objects from a set of $n$ distinct objects without regard to order is given by $^nC_r = \frac{n!}{r!(n-r)!}$.
• Permutations ($n!$): The number of ways to arrange $n$ distinct objects in a sequence is given by $n!$.
• Inclusion-Exclusion Principle: A counting technique to find the number of elements in the union of multiple sets.

Step-by-Step Solution

Step 1: Calculate the total number of ways to place the letters without any restrictions, except that at most one letter per box is allowed.

We have 8 boxes and 5 distinct letters. We need to choose 5 boxes out of 8 to place the letters. This can be done in $^8C_5$ ways. Then, we can arrange the 5 distinct letters in the selected 5 boxes in $5!$ ways. Therefore, the total number of ways to place the letters without any restrictions (except one letter per box) is: $^8C_5 \times 5! = \frac{8!}{5!3!} \times 5! = \frac{8!}{3!} = 8 \times 7 \times 6 \times 5 \times 4 = 6720$

Step 2: Calculate the number of ways where row 1 is empty.

If row 1 is empty, we have 5 letters to place in the remaining 5 boxes (3 in row 2 and 2 in row 3). We choose 5 boxes from these 5 boxes and arrange the letters. This can be done in $^5C_5 \times 5! = 1 \times 5! = 120$ ways.

Step 3: Calculate the number of ways where row 2 is empty.

If row 2 is empty, we have 5 letters to place in the remaining 5 boxes (3 in row 1 and 2 in row 3). We choose 5 boxes from these 5 boxes and arrange the letters. This can be done in $^5C_5 \times 5! = 1 \times 5! = 120$ ways.

Step 4: Calculate the number of ways where row 3 is empty.

If row 3 is empty, we have 5 letters to place in the remaining 6 boxes (3 in row 1 and 3 in row 2). We choose 5 boxes from these 6 boxes and arrange the letters. This can be done in $^6C_5 \times 5! = 6 \times 5! = 6 \times 120 = 720$ ways.

Step 5: Calculate the number of ways where both row 1 and row 2 are empty.

If both row 1 and row 2 are empty, then all 5 letters must be placed in the 2 boxes of row 3. This is impossible since we only have 2 boxes. So, the number of ways is 0.

Step 6: Calculate the number of ways where both row 1 and row 3 are empty.

If both row 1 and row 3 are empty, then all 5 letters must be placed in the 3 boxes of row 2. This is impossible since we only have 3 boxes. So, the number of ways is 0.

Step 7: Calculate the number of ways where both row 2 and row 3 are empty.

If both row 2 and row 3 are empty, then all 5 letters must be placed in the 3 boxes of row 1. This is impossible since we only have 3 boxes. So, the number of ways is 0.

Step 8: Apply the Inclusion-Exclusion Principle.

Let $N(R_i)$ be the number of arrangements where row $i$ is empty. We want to find the number of arrangements where no row is empty. By the Inclusion-Exclusion Principle, this is:

Total arrangements - (Arrangements with row 1 empty + Arrangements with row 2 empty + Arrangements with row 3 empty) + (Arrangements with row 1 and 2 empty + Arrangements with row 1 and 3 empty + Arrangements with row 2 and 3 empty) - (Arrangements with row 1, 2, and 3 empty).

So, the number of ways is: $6720 - (120 + 120 + 720) + (0 + 0 + 0) - 0 = 6720 - 960 = 5760$

Step 9: Check if the answer matches the possible answers

5760 is one of the possible answers. However, we are told the correct answer is 5880. This indicates an error in the problem itself, or in the provided "Correct Answer." Assuming the logic here is sound, 5760 is the correct derivation.

Common Mistakes & Tips

• Be careful to distinguish between permutations and combinations. Permutations are used when the order matters, while combinations are used when the order does not matter.
• When applying the Inclusion-Exclusion Principle, make sure you consider all possible combinations of empty rows.
• Double-check your calculations to avoid arithmetic errors.

Summary

We calculated the total number of ways to place the 5 distinct letters into the 8 boxes without any row being empty. We used the Inclusion-Exclusion Principle to subtract the cases where at least one row is empty. After calculation, we find that the number of ways is 5760. However, the given correct answer is 5880, which suggests an error in the problem statement or answer key. Assuming the logic is sound, the derivation yields 5760.

Final Answer

The final answer is \boxed{5760}, which corresponds to option (D).