Number of even numbers with sum of digits less than 12 How many positive three digit even numbers lesser than 700 are there with all three different digits and where the sum of the three digits is 12 or less ?  

Am I supposed to write solve the expression for each of the 12 cases like solution for 
a + b + C=12, 11, 10...1 and then add them together? 

Is there a better approach to this problem?
I would just enter the numbers 2 through 698 into a spreadsheet, use the mod function to extract the digits, and if statements to determine whether the digits are different and sum to less than 12.  Then ask it to count the results.  The copy function means you only write the things once.
9+3+0
9+2+1
8+4+0
8+3+1
8+2+2 (error)
7+5+0
7+4+1
7+3+2
7+2+2 (error)
6+5+1
6+4+2
6+3+3 (error)
5+4+3


Each can be permuted ABC, ACB, BAC, BCA, CAB, CBA, so 13x6 if I counted right.

[Oops, I failed to eliminate the numbers above 700.  We'll leave that as an exercise for the student.  Easily done by breaking the above list into two ranges.]

[Gotta learn to read more carefully!  Sum of digits LESS than 12!  Oh well...]

[I suspect the above scheme would work, though, with a little adjustment.]