I don't see any easy way to do this. You have to go back to the equation

and check (for each value of S) whether there is a solution in which a, b, c and d are distinct digits. The smallest possible value of S is 83 (obtained when a=1, b=2, c=3, d=4) and the largest possible value is 357 (a=9, b=8, c=7, d=6). The only values of S for which no solution is possible are going to occur when S is fairly close to those extreme values. When S is well inside the interval [83,357] (for example, when S=200, as in the original problem) there is enough flexibility in the choice of a,b,c,d to ensure that there will always be at least one solution.

Notice also that if (a,b,c,d) is a solution for S, then (10–a,10–b,10–c,10–d) will be a solution for 440–S. So the values of S for which there is no solution will come in pairs adding up to 440. I think that there are 22 such values, namely

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