The sum of all numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time, is equal to:

Question

Accepted Answer

To determine the sum of all numbers that can be formed with the digits 2, 3, 4, and 5 taken all at a time, we need to consider all possible permutations of these four digits. Since there are 4 digits, the total number of permutations is 4! (factorial of 4), which equals 24.

Each digit will appear in each decimal place (thousands, hundreds, tens, and units) an equal number of times across all permutations. Specifically, each digit will appear in each place 6 times (since 24 permutations divided by 4 places equals 6).

Let's calculate the contribution of each digit to the total sum:

1. **Thousands place**: Each digit appears 6 times in the thousands place. Therefore, the contribution from the thousands place is:
   $$
   6 \times (2000 + 3000 + 4000 + 5000) = 6 \times 14000 = 84000
   $$

2. **Hundreds place**: Similarly, each digit appears 6 times in the hundreds place. The contribution from the hundreds place is:
   $$
   6 \times (200 + 300 + 400 + 500) = 6 \times 1400 = 8400
   $$

3. **Tens place**: Each digit appears 6 times in the tens place. The contribution from the tens place is:
   $$
   6 \times (20 + 30 + 40 + 50) = 6 \times 140 = 840
   $$

4. **Units place**: Each digit appears 6 times in the units place. The contribution from the units place is:
   $$
   6 \times (2 + 3 + 4 + 5) = 6 \times 14 = 84
   $$

Adding all these contributions together gives the total sum:
$$
84000 + 8400 + 840 + 84 = 93324
$$

Therefore, the sum of all numbers that can be formed with the digits 2, 3, 4, and 5 is 93324, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (93328)**: This option is incorrect because the calculated sum is 93324, not 93328. The discrepancy arises from incorrect addition or miscalculation in the contribution of digits.

- **Option 3 (92324)**: This option is incorrect because it underestimates the total sum. The correct calculation shows that the sum is 93324, not 92324.

- **Option 4 (92328)**: This option is incorrect for similar reasons as Option 3. The calculated sum is 93324, not 92328, indicating an error in the calculation process.

In conclusion, Option 1 is the correct answer as it accurately reflects the sum of all permutations of the digits 2, 3, 4, and 5.