No of ways we can form a 4 digit no= 5*6*6*3=540

since there can be 3 digits at ones place, no. of times each digit will appear=(540/3)=180
sum of digits at ones place =(0+2+4)*180=1080

there can be 6 digits at tens place, no. of times each digit will appear=(540/6)=90
sum of digits at tens place=(0+1+2+3+4+5)*90*10=13500

similarly there can be 6 digits at hundreds place again
sum of digits at hundreds place=(0+1+2+3+4+5)*90*100=135000

there can be 5 digits at thousands place, no. of times each digit will appear=(540/5)=108
sum of digits=(1+2+3+4+5)*108*1000=1620000

sum of all the numbers=sum of digits at all the places=(1620000+135000+13500+1080)=1769580

Solution:
The given digits are 0, 1, 2, 3, 4, and 5. We have to form even four-digit numbers using these digits. For a number to be even, its unit place digit must be either 0, 2, or 4.

- Case 1: Unit place digit is 0
If the unit place digit is 0, then the thousands place digit can be filled in 5 ways (as 0 cannot be the thousands place digit). For each choice of the thousands place digit, the hundreds place digit can be filled in 5 ways (as 0 is not allowed and one digit has already been used). Similarly, the tens place digit can be filled in 4 ways (as two digits have already been used) and the unit place digit is fixed as 0. Hence, the total number of even four-digit numbers with 0 in the unit place is 5 x 5 x 4 = 100.

- Case 2: Unit place digit is 2 or 4
If the unit place digit is 2 or 4, then the thousands place digit can be filled in 6 ways (as 0 can also be used now). For each choice of the thousands place digit, the hundreds place digit can be filled in 5 ways (as one digit has already been used). Similarly, the tens place digit can be filled in 4 ways (as two digits have already been used) and the unit place digit can be filled in 2 ways (as either 2 or 4 can be used). Hence, the total number of even four-digit numbers with 2 or 4 in the unit place is 6 x 5 x 4 x 2 = 240.

Therefore, the total sum of all even four-digit numbers that can be formed using the digits 0, 1, 2, 3, 4, and 5 is (100 x 0) + (240 x 2) + (240 x 4) = 1760 + 960 = 2720.

However, we have counted each digit (0, 2, and 4) in the unit place twice (once for each of the other two digits). Hence, we need to divide the sum by 2 to get the actual sum of even four-digit numbers. Therefore, the actual sum is 2720/2 = 1360.

Hence, the correct option is 3. 1769580.