number of unequal non negative(including 0) integral solutions of a+b+c+d=20(the answer is 552)

It is a very cumbersome question and beyond the scope of CAT or any man management exam for that matter.
Yet I will help you with the framework and you can solve the question if you want.
Find the total integral solutions which is given by n+r-1 Cr-1, 23C3 =1771
Now remove the case where all 4 are equal, Now remove the cases where 3 are equal.
Similarly, remove the cases where any two are equal. This is the only way and there will be multiple cases in the case where any two are equal.

So I would suggest you not to waste time on such questions and solve materials which are relevent to CAT.
Thanks