Question:
The number of ways of distributing \(500\) dissimilar boxes equally among \(50\) persons is
The number of ways of distributing \(500\) dissimilar boxes equally among \(50\) persons is
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When \(n\) distinct objects are distributed equally among distinct groups, use multinomial distribution:
\[
\frac{n!}{(r!)^k}
\]
where each group gets \(r\) objects and there are \(k\) groups.
Updated On: Jun 25, 2026
- \(\dfrac{500!}{(10!)^{50}\,50!}\)
- \(\dfrac{500!}{(50!)^{10}\,10!}\)
- \(\dfrac{500!}{(50!)^{10}}\)
- \(\dfrac{500!}{(10!)^{50}}\)
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The Correct Option is D
Solution and Explanation
Step 1: Understand the distribution condition.
There are \[ 500 \] dissimilar boxes. They are to be distributed equally among \[ 50 \] persons. So each person must get \[ \frac{500}{50}=10 \] boxes.
Step 2: Since persons are distinct.
The \(50\) persons are considered distinct.
So, we do not divide by \(50!\).
Step 3: Distribute boxes equally.
The first person can receive \(10\) boxes, the second person can receive \(10\) boxes, and so on.
The total number of ways is \[ \frac{500!}{10!10!10!\cdots 10!} \] Since there are \(50\) persons, \(10!\) appears \(50\) times.
Hence, \[ \frac{500!}{(10!)^{50}} \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{500!}{(10!)^{50}}} \]
There are \[ 500 \] dissimilar boxes. They are to be distributed equally among \[ 50 \] persons. So each person must get \[ \frac{500}{50}=10 \] boxes.
Step 2: Since persons are distinct.
The \(50\) persons are considered distinct.
So, we do not divide by \(50!\).
Step 3: Distribute boxes equally.
The first person can receive \(10\) boxes, the second person can receive \(10\) boxes, and so on.
The total number of ways is \[ \frac{500!}{10!10!10!\cdots 10!} \] Since there are \(50\) persons, \(10!\) appears \(50\) times.
Hence, \[ \frac{500!}{(10!)^{50}} \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{500!}{(10!)^{50}}} \]
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