Question

The number of ways of distributing \( 500 \) dissimilar boxes equally among \( 50 \) persons is

• A. \( \frac{500!}{(10!)^{50} \cdot 50!} \)
• B. \( \frac{500!}{(50!)^{10} \cdot 10!} \)
• C. \( \frac{500!}{(50!)^{10}} \)
• D. \( \frac{500!}{(10!)^{50}} \)

Hint:

When distributing distinct objects among several groups with equal items per group, the formula \(\frac{\text{Total factorial}}{(\text{Group factorial})^{\text{Number of groups}}}\) gives the number of possible arrangements.

Answer 1

The Correct Option is D

To distribute \( 500 \) distinct boxes equally among \( 50 \) individuals, such that each person receives \( 10 \) boxes, the number of possible distributions is calculated using the formula: \[ \frac{500!}{(10!)^{50}}. \] This formula arises because: - The \( 500! \) term represents the total permutations of all \( 500 \) distinct boxes. - For each of the \( 50 \) individuals, the \( 10 \) boxes they receive can be internally arranged in \( 10! \) ways. - To correct for overcounting due to the identical distribution patterns among individuals, we divide by \( (10!)^{50} \), accounting for the \( 10! \) arrangements for each of the \( 50 \) recipients.

Final Answer: \[ \boxed{\frac{500!}{(10!)^{50}}} \]