Question:
The number of ways of dividing \(200\) dissimilar things into \(10\) groups each containing \(20\) elements is:
The number of ways of dividing \(200\) dissimilar things into \(10\) groups each containing \(20\) elements is:
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When dividing distinct objects into equal-sized unlabelled groups, first divide by the factorials of the group sizes and then divide by the factorial of the number of groups.
Updated On: Jun 18, 2026
- \[ \frac{200!}{(20!)^{10}\,10!} \]
- \[ \frac{200!}{(10!)^{10}\,20!} \]
- \[ \frac{200!}{(20!)^{10}\,10!} \]
- \[ \frac{200!}{(10!)^{20}\,20!} \]
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The Correct Option is C
Solution and Explanation
Step 1: Arrange the \(200\) distinct objects into \(10\) groups of \(20\) each.
Since all \(200\) objects are distinct, the number of ways to distribute them into ordered groups of sizes \[ 20,20,\ldots,20 \] is \[ \frac{200!}{(20!)^{10}}. \] The factor \((20!)^{10}\) removes the arrangements within each group.
Step 2: Adjust for identical groups.
The groups are not labelled.
Any permutation of the \(10\) groups gives the same division.
The number of ways to arrange \(10\) groups is \[ 10!. \] Therefore, we divide by \(10!\).
Step 3: Obtain the required number of divisions.
Hence, \[ \text{Number of ways} = \frac{200!}{(20!)^{10}\,10!}. \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{200!}{(20!)^{10}\,10!}} \]
Since all \(200\) objects are distinct, the number of ways to distribute them into ordered groups of sizes \[ 20,20,\ldots,20 \] is \[ \frac{200!}{(20!)^{10}}. \] The factor \((20!)^{10}\) removes the arrangements within each group.
Step 2: Adjust for identical groups.
The groups are not labelled.
Any permutation of the \(10\) groups gives the same division.
The number of ways to arrange \(10\) groups is \[ 10!. \] Therefore, we divide by \(10!\).
Step 3: Obtain the required number of divisions.
Hence, \[ \text{Number of ways} = \frac{200!}{(20!)^{10}\,10!}. \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{200!}{(20!)^{10}\,10!}} \]
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