To find the sum of all 4-digit numbers that can be formed using the digits 1, 2, 3, and 4 with no repetition, we need to consider all possible permutations of these digits.

We can solve this problem by breaking it down into smaller steps:

Step 1: Determine the total number of permutations
Since we have 4 digits and no repetition is allowed, the number of permutations can be calculated as 4 factorial (4!):
4! = 4 x 3 x 2 x 1 = 24

Step 2: Find the sum of all possible thousands digits
Since the thousands digit can be any of the four given digits (1, 2, 3, or 4), we need to find the sum of these digits and then multiply it by the number of permutations.
Sum of the digits = 1 + 2 + 3 + 4 = 10
Sum of the thousands digits = 10 x 24 = 240

Step 3: Find the sum of all possible hundreds digits
Similarly, the sum of the hundreds digits can be found by multiplying the sum of the digits (1 + 2 + 3 + 4) by the number of permutations.
Sum of the hundreds digits = 10 x 24 = 240

Step 4: Find the sum of all possible tens digits
Again, the sum of the tens digits can be found by multiplying the sum of the digits (1 + 2 + 3 + 4) by the number of permutations.
Sum of the tens digits = 10 x 24 = 240

Step 5: Find the sum of all possible units digits
The sum of the units digits can be found using the same method as above.
Sum of the units digits = 10 x 24 = 240

Step 6: Calculate the final sum
To find the sum of all 4-digit numbers, we need to add up the sums of the thousands, hundreds, tens, and units digits.
Final sum = Sum of thousands digits + Sum of hundreds digits + Sum of tens digits + Sum of units digits
Final sum = 240 + 240 + 240 + 240 = 960

Therefore, the correct answer is option A) 66660.