Let the set S = {2, 4, 8, 16, ldots, 512} be | vedclass

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(A) The set $S = \{2^1, 2^2, 2^3, \ldots, 2^9\}$ contains $9$ elements.We need to partition these $9$ elements into $3$ sets $A, B, C$ each containing

Vedclass · Accepted Answer

$1680$

Vedclass · Answer

(A) The set $S = \{2^1, 2^2, 2^3, \ldots, 2^9\}$ contains $9$ elements.We need to partition these $9$ elements into $3$ sets $A, B, C$ each containing $3$ elements.The number of ways to divide $9$ distinct objects into $3$ groups of $3$ is given by the multinomial coefficient:$\frac{9!}{3! 3! 3! 3!}$Since the sets $A, B, C$ are distinct (labeled),we multiply by $3!$ to assign the groups to the sets $A, B, C$:$	ext{Number of ways} = \frac{9!}{3! 3! 3! 3!} 	imes 3! = \frac{9!}{3! 3! 3!} = \frac{362880}{6 	imes 6 	imes 6} = \frac{362880}{216} = 1680$.

Let the set $S = \{2, 4, 8, 16, \ldots, 512\}$ be partitioned into $3$ sets $A, B, C$ with an equal number of elements such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = A \cap C = \phi$. The number of such possible partitions of $S$ is equal to:

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