Question

If all possible 4-digit numbers are formed by choosing 4 different digits from the given digits $ 1, 2, 3, 5, 8 $, then the sum of all such 4-digit numbers is:

• A. $199980$
• B. $999990$
• C. $506616$
• D. $479952$

Hint:

When calculating the sum of all permutations of selected digits, calculate how many times each digit contributes to each place value and sum accordingly. Make sure to adjust for the exact number of digits used per number.

Answer 1

The Correct Option is C

Step 1: Determine Total Number of Permutations
We have 5 distinct digits \(\{1, 2, 3, 5, 8\}\) and need to form 4-digit numbers using 4 different digits. The number of such permutations is: \[ P(5,4) = 5 \times 4 \times 3 \times 2 = 120 \text{ numbers}. \] Step 2: Calculate Frequency of Each Digit in Each Place
For any given digit (say \(1\)), it will appear in: Thousands place: \(P(4,3) = 24\) times
Hundreds place: \(P(4,3) = 24\) times
Tens place: \(P(4,3) = 24\) times
Units place: \(P(4,3) = 24\) times
This symmetry holds for all digits. 
Step 3: Compute Sum for Each Place Value
The sum contributed by each digit in each place is: \[ \text{Sum per digit} = \text{Digit} \times 24 \times \text{Place value}. \] Total sum: \[ \sum_{\text{all digits}} \sum_{\text{all places}} (\text{Digit} \times 24 \times \text{Place value}). \] Breaking it down by place values: 


Step 4: Calculate Total Sum
Adding all place contributions: \[ 456000 + 45600 + 4560 + 456 = 506616. \] Conclusion
The sum of all possible 4-digit numbers formed is \(\boxed{506616}\).