How to find the number of possible combinations of abcd given a+b+c+d=100?

Question

Mathematics

Find the number of possible combinations of abcd under the condition a+b+c+d=100 (abcd are non-negative integers).
Of course, it is easy to calculate using any programming language, but it became interesting how to deal with this issue from the point of view of mathematics.

Answer 1

If zero parts are allowed in compositions of number n of length k, then the number of such compositions will be equal to .
en.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%...
en.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD %D0%BE%D0%...

Answer 2

In short, follow @friogenn 's link .
In more detail, to solve such a problem, it is convenient to represent the number 100 as a hundred units that need to be scattered over 4 baskets without returns. If zeros are not allowed in the decomposition, then throw one into each basket and distribute the remaining 96.
In other words, place 4 "partitions" in 99 (if empty groups are not allowed) or 101 (if allowed) positions. After that, it is necessary to subtract duplicate results (basket permutations).