Solutions to equations involving a sum of integer valued variables. About (a) How many solutions are there to the equation where each is a non-negative integer

I think your question is missing something,i tried to explain you by examples.

Explanation:

1)

n = 37, k = 6.

Thus, the answer is (37 + 6 − 1)C6-1 = 42C5

=> C(42,5) = 850668

Where C(n,r) is the combination

2)

n = 37

Since xi >= 2 that means x1 >= 2 , x2 >= 2, x3 >= 2 , x4 >= 2, x5 >= 2, x6 >= 2

Then, we distribute the remaining 37 − 2 − 2 − 2 − 2 − 2 - 2 = 25

Thus, the answer is (25 + 6 − 1)C6-1 = 30C5

=> C(30,5) = 142506

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