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