Question:
\( S = \{1,2,3,\dots,20\} \) is to be partitioned into four sets \( A, B, C, D \) of equal size. The number of ways is
\( S = \{1,2,3,\dots,20\} \) is to be partitioned into four sets \( A, B, C, D \) of equal size. The number of ways is
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Partition into equal labeled sets → divide by factorial of group sizes only.
Updated On: Apr 23, 2026
- $\dfrac{20!}{4!\cdot 5!}$
- $\dfrac{20!}{4^5}$
- $\dfrac{20!}{(5!)^4}$
- $\dfrac{20!}{(4!)^5}$
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The Correct Option is C
Solution and Explanation
Concept:
Partition of set into equal groups
Step 1: Total elements = 20, each set has 5 elements.
Step 2: Arrange all elements: 20!
Step 3: Divide by $(5!)^4$ (internal arrangements of each set).
Step 4: Since sets are distinct (A, B, C, D), no further division.
Conclusion:
Number of ways = $\dfrac{20!}{(5!)^4}$
Step 1: Total elements = 20, each set has 5 elements.
Step 2: Arrange all elements: 20!
Step 3: Divide by $(5!)^4$ (internal arrangements of each set).
Step 4: Since sets are distinct (A, B, C, D), no further division.
Conclusion:
Number of ways = $\dfrac{20!}{(5!)^4}$
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