Determine how many integer solutions there are to a+b+c+d=19 if
It is easy to do with generating functions.
I will start you off with inclusion/exclusion.
The number of ways for condition a to happen is .
The number of ways for conditions c to happen is .
The number of ways for conditions a & c to happen together is .
Now to finish you have to figure for each of the other conditions in all possible combinations.
With generating functions we expand .
The coefficient of will be the answer.