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

Answer: (A)

1680

Answer 2

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.