In an entrance examination, there are two papers of 50 marks and one paper of 100 marks. A candidate will qualify to take admission if he secures at least 150 marks in aggregate, securing at least 50 marks in the last paper. In how different ways can he secure marks in the three papers to qualify?

The above question is similar to finding all the non-negative integral solutions to the equation:

a + b + d = z

Where

a,b belong to [0,50]

d belongs to [50,100]

z belongs to [150,200]

Now how to solve after that....help required to get the answer. (The biggest problem is that z itself is variable)

Thanks in advance!!

UPDATE:

The answer is coming out to be a combo of three combinatorial series....need help after this:

```
       (102 C 100) + (103 C 101) + ..... + (152 C 150)

- 3×[ (51 C 49) + (52 C 50) + .....  +(101 C 99)] 

- 3×(51 C 48) =

```