Question

Let the set $S = \{2, 4, 8, 16, ..., 512\}$ 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:

• A. 1680
• B. 1520
• C. 1710
• D. 1640

Answer 1

The Correct Option is A

The set \( S = \{ 2, 2^2, 2^3, \ldots, 2^9 \} \) contains 9 elements. To partition \( S \) into 3 subsets \( A, B, C \) of equal size, each subset must have exactly 3 elements.

The number of ways to partition the set can be calculated using the formula:

\[ \text{Number of partitions} = \frac{9!}{(3!3!3!)} \times 3!. \]

Expanding this expression:

\[ \text{Number of partitions} = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4}{6 \times 6} \times 6 = 1680. \]

Therefore, the maximum number of such possible partitions of \( S \) is 1680.