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

The task is to distribute \( 500 \) distinct boxes evenly among \( 50 \) individuals, with each person receiving \( 10 \) boxes. To determine the number of ways this can be done, we use the following formula: \[ \frac{500!}{(10!)^{50}}. \] Explanation: - The total number of ways to arrange all the \( 500 \) distinct boxes is \( 500! \). - Each of the \( 50 \) individuals receives \( 10 \) boxes, and the internal arrangement of the boxes for each individual can be done in \( 10! \) ways. - Since there are \( 50 \) people, we need to divide by \( (10!)^{50} \) to account for the repeated arrangements among the boxes given to each person. Final Answer: \[ \boxed{\frac{500!}{(10!)^{50}}} \]