Question:
The set $S=\left\{1, 2, 3, \dots, 12\right\}$ is to be partitioned into three sets $A, B, C$ of equal size. Thus, $A\cup B\cup C=S, A\cap B = B\cap C = A \cap C=\phi.$ The number of ways to partition $S$ is
The set $S=\left\{1, 2, 3, \dots, 12\right\}$ is to be partitioned into three sets $A, B, C$ of equal size. Thus, $A\cup B\cup C=S, A\cap B = B\cap C = A \cap C=\phi.$ The number of ways to partition $S$ is
Updated On: Jul 5, 2022
- $\frac{12!}{3!\left(4!\right)^{3}}$
- $\frac{12!}{3!\left(3!\right)^{4}}$
- $\frac{12!}{\left(4!\right)^{3}}$
- $\frac{12!}{\left(3!\right)^{4}}$
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The Correct Option is C
Solution and Explanation
Number of ways is $^{12}C_{4}\times^{8}C_{4}\times^{4}C_{4}$
$=\frac{12!}{\left(4!\right)^{3}}$
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Concepts Used:
Permutations and Combinations
Permutation:
Permutation is the method or the act of arranging members of a set into an order or a sequence.
- In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
- A permutation is used in many events of daily life. It is used for a list of data where the data order matters.
Combination:
Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.
- Combination refers to the combination of about n things taken k at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.