Determine how many integer solutions there are to a+b+c+d=19 if
0<=a<=5,
0<=b<=6,
3<=c<=7,
3<=d<=8
This is an extremely tedious problem to do with inclusion/exclusion!
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.