Answer:

To find the sum of all 4-digit numbers that can be formed using the numbers 1, 3, 5, and 7, we need to consider all possible permutations of these digits.

Total Number of Combinations:
To determine the total number of combinations, we can use the formula for permutations. Since we are selecting 4 digits out of 4 available digits, the total number of combinations is given by:
4P4 = 4! / (4 - 4)! = 4! / 0! = 4 x 3 x 2 x 1 = 24

Finding the Sum:
To find the sum of all these 4-digit numbers, we can look at each digit's position and calculate the sum accordingly.

Thousands Digit:
The thousands digit can take any of the four available digits: 1, 3, 5, or 7. Each digit will appear in this position six times since there are six combinations possible with the remaining three digits. So, the sum of the thousands digit will be:
(1 + 3 + 5 + 7) x 6 = 16 x 6 = 96

Hundreds Digit:
The hundreds digit can also take any of the four available digits. However, each digit will appear in this position four times since there are only four combinations possible with the remaining two digits. So, the sum of the hundreds digit will be:
(1 + 3 + 5 + 7) x 4 = 16 x 4 = 64

Tens and Units Digits:
Similarly, the tens and units digits can each take any of the four available digits. However, each digit will appear in these positions twice since there are only two combinations possible with the remaining one digit. So, the sum of both the tens and units digits will be:
(1 + 3 + 5 + 7) x 2 = 16 x 2 = 32

Total Sum:
To get the total sum, we need to sum up the values of the thousands, hundreds, tens, and units digits:
96 (thousands digit) + 64 (hundreds digit) + 32 (tens digit) + 32 (units digit) = 224

Therefore, the sum of all 4-digit numbers that can be formed using the digits 1, 3, 5, and 7 is 224.