Step 1: Understanding the Concept:
Digits: 1, 4, 2, 1. Total distinct permutations $= \frac{4!}{2!} = 12$. Step 2: Key Formula or Approach:
Sum of all numbers $= (\text{Sum of unique digits}) \times (\text{Permutations per digit in a place}) \times (1111)$.
Mean $= \frac{\text{Total Sum}}{\text{Total Permutations}}$. Step 3: Detailed Explanation:
Sum of digits in any place:
Digit 4 appears $3!/2! = 3$ times.
Digit 2 appears $3!/2! = 3$ times.
Digit 1 appears $3! = 6$ times.
Sum per place $= (4 \times 3) + (2 \times 3) + (1 \times 6) = 12 + 6 + 6 = 24$.
Total Sum $= 24(10^3) + 24(10^2) + 24(10^1) + 24(10^0) = 24 \times 1111 = 26664$.
Mean $= \frac{26664}{12} = 2222$. Step 4: Final Answer:
The arithmetic mean is 2222.