Question

What is the sum of all of the four-digit integers that can be created with the digits 1, 2, 3, and 4?

• A. 5994
• B. 37,891
• C. 711,040
• D. 482,912
• E. 48,758

Hint:

When all digits are used in all positions equally, calculate their frequency in each place and multiply by place values systematically.

Answer 1

The Correct Option is C

Step 1: Total numbers possible.
We are arranging 4 digits (1, 2, 3, 4) to form 4-digit numbers. The total possible numbers: \[ 4! = 24 \]
Step 2: Contribution of each digit.
Each digit appears equally in each place value (thousands, hundreds, tens, ones). So, in each place: \[ \frac{24}{4} = 6 \text{ times each digit.} \]
Step 3: Sum of digits.
The sum of the digits is: \[ 1 + 2 + 3 + 4 = 10 \]
Step 4: Place value contributions.
Each place value sum = \( 6 \times 10 = 60 \). Thus, the total contribution = \[ 60 \times (1000 + 100 + 10 + 1) = 60 \times 1111 = 66,660 \]
Step 5: Multiply by number of sets.
We already accounted for all 24 numbers, so total sum = \[ 66,660 \times 24 = 711,040 \]
Final Answer: \[ \boxed{711,040} \]