Smallest possible number possible for abcd where a+b+c = d is 6 and the greatest is 6.

1+2+x:

smallest value x can be is 3, while the greatest is 6.

\(6-3+1=4\) There are 4 numbers for x which when added to 1 and 2, will equal to a digit.

4 NUMBERS

1+3+x:

x can only be 4 or 5. 2 is already covered in the previous part. if x is 4 we get 8 and if x is 5 we get 9.

2 NUMBERS

2+3+x:

2+3 = 5, which means 1 ≤ x ≤ 4. However, 1 has already been covered, 2 and 3 won't work as they would not be distinct, which leaves only 4.

1 NUMBER

Total: \(4+2+1=7\) so there are 7 total numbers which work. However, the 3 digits before d (abc), can be arranged in \(3!\) ways.

\(7*3! = 42\)

Answer E