Question

The digits of the number 235 are permutated to form 6 numbers (including 235). What is the sum of the resulting 6 numbers?

• A. 2320
• B. 2120
• C. 2220
• D. 2310

Hint:

In digit permutation sums, each digit repeats $(n-1)!$ times in each place.

Answer 1

The Correct Option is C

To solve the problem of finding the sum of numbers formed by permutating the digits of 235, let's proceed step-by-step.

Identify all permutations of the digits 2, 3, and 5:

• 235
• 253
• 325
• 352
• 523
• 532

Calculate the sum of these numbers:

\(235 + 253 + 325 + 352 + 523 + 532\)

Let's add them step by step:

• \(235 + 253 = 488\)
• \(488 + 325 = 813\)
• \(813 + 352 = 1165\)
• \(1165 + 523 = 1688\)
• \(1688 + 532 = 2220\)

Verification:

This can also be systematically verified by considering the positions and the frequency of each digit occupying each position (hundreds, tens, units).

• Each digit (2, 3, 5) appears twice in each position (hundreds, tens, units) across all permutations.
• The contribution to the total sum by each position:
  Hundreds: \(2 \times (200 + 300 + 500) = 2000\)
  Tens: \(2 \times (20 + 30 + 50) = 200\)
  Units: \(2 \times (2 + 3 + 5) = 20\)

Conclusion:

The sum of the resulting 6 numbers formed by permutating the digits of 235 is 2220.

Thus, the correct answer is: 2220.