Distributing marbles

How many ways are there to distribute 10 identical marbles in 4 identical bowls, with at most 6 marbles in any one bowl ?

This problem can be reformulated as solving an equation with 4 unknowns, using non-negative integer solutions :

First, with no restriction, the total number of ways is :

Among which there are some which violate the given conditions, i.e., having 1 bowl containing more than 6 marbles :

So the total number of possible quartets is 286 – 80 = 206.

Another way to solve this problem is to work with the polynomial :

Look for the terms in x¹⁰ and extract their coefficients :

The first method is called inclusion-exclusion. The second method is by generating functions.