The Sum of All Four-Digit Numbers Using Digits 0, 1, 2, and 3

To find the sum of all four-digit numbers that can be formed using the digits 0, 1, 2, and 3, we need to consider the possible combinations. The four-digit numbers can be formed by using these digits as thousands, hundreds, tens, and units places.

Step 1: Find all the possible combinations

To find the total number of combinations, we can use the concept of permutations. Since we have four digits to choose from and four positions to fill, the total number of combinations is given by 4P4 = 4! = 4 x 3 x 2 x 1 = 24.

Step 2: Determine the sum of each digit at different places

The sum of each digit at different places remains the same for all the combinations. Let's calculate the sum of each digit at each place:

Thousands place: The digit 0, 1, 2, and 3 each appears 6 times at this place since there are 24 combinations in total. So, the sum of the digits at the thousands place is (0 + 1 + 2 + 3) x 6 = 24.

Hundreds place: The digit 0, 1, 2, and 3 each appears 6 times at this place. So, the sum of the digits at the hundreds place is (0 + 1 + 2 + 3) x 6 = 24.

Tens place: The digit 0, 1, 2, and 3 each appears 6 times at this place. So, the sum of the digits at the tens place is (0 + 1 + 2 + 3) x 6 = 24.

Units place: The digit 0, 1, 2, and 3 each appears 6 times at this place. So, the sum of the digits at the units place is (0 + 1 + 2 + 3) x 6 = 24.

Step 3: Calculate the total sum

To find the sum of all the numbers, we need to add up the sum of each digit at different places. Therefore, the total sum is 24 + 24 + 24 + 24 = 96.

Conclusion

The sum of all the numbers of four different digits that can be made by using the digits 0, 1, 2, and 3 is 96. None of the given options (a, b, or d) matches the correct answer. The correct answer is option 'C' (38664).