If we add 2 to each of them we get
a + b + c + d = 18
and each is greater or equal to 1.
The number of solutions is the same.
Imagine 18 counters in a row with spaces between them
o _ o _ o _ o _ ... o _ o _ o _ o
There are 17 gaps between then, and we must choose 3 of them
to divide the 18 counters into 4 piles.
So there are 17c3 = 680 ways to do that
resulting in the four numbers a, b, c, d adding up to 18.
Then subtract 2 from each number to transform it back to the original problem,
but the number of ways remains the same.
This is called a "Stars and Bars" method,
as it is a way to divide N stars into K groups by placing (K-1) bars between them
* * * * | * * * | * | *
by requiring each group to have at least 1 in it,
we avoid the messiness of worrying about spaces with more than one bar,
and even more messiness if we allow negatives.
Hence the transformation to all components >= 1.