Sets, Relations & Functions
Sets — Partitioning into Equal Subsets
nta_pyq_2024_apr
Grade 11
Question:
Let the set $S=\{2,4,8,16,\ldots,512\}$ be partitioned into 3 sets $A,B,C$ with equal number of elements such that $A\cup B\cup C=S$ and $A\cap B=B\cap C=A\cap C=\phi$. The maximum number of such possible partitions of $S$ is equal to:
Step-by-Step Solution
Key Concept: $S=\{2^1,2^2,\ldots,2^9\}$ has 9 elements. Each of $A,B,C$ must have exactly 3 elements. Number of ways to partition 9 distinct elements into 3 unordered groups of 3 is $\dfrac{9!}{3!\cdot3!\cdot3!\cdot3!}$, but since $A,B,C$ are labelled (distinct sets), the count is $\dfrac{9!}{3!\cdot3!\cdot3!}$.
$|S|=9$, each set has 3 elements. Labelled partition count $=\dfrac{9!}{3!\,3!\,3!}=1680$.
Correct Answer: 1