Permutations and Combinations
Fundamental Principle of Counting
Premium Question
Grade 11
Question:
The number of 6-digit numbers formed using digits $\{1, 2, 3, 4\}$ (with repetition) that are divisible by 4 is:
(1) $512$
(2) $1024$
(3) $2048$
(4) $256$
Step-by-Step Solution
Key Concept: A number is divisible by 4 iff its last two digits form a number divisible by 4. From $\{1, 2, 3, 4\}$: valid endings are $12, 24, 32, 44$ — exactly 4 choices. First 4 digits: $4^4 = 256$ each. Total $= 4 \times 256 = 1024$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)