Sir, what is the difference between non-negative integral solutions and positive integral solutions in permutations and combinations ???

Here,  one thing you should remember is that, non negative means all the numbers which are not negative. Therefore 0 and other positive numbers combined are known as non negative numbers.  Also, positive means,  all the positive integers i.e all integers greater than 0.

Coming to permutations and combinations, if the question is something like to find non negative integral solutions for distributing boxes, then you should consider the solutions where each will get 0 or more boxes.  And if it for positive integral solutions  then you should consider the solutions ehe each will get 1 or more boxes.