How many integer solutions (x1; x2; x3) are there to the equation :-
x1 + x2 + x3 = 25
with all xi satisfying 0 < xi < 10?
Maybe not the most elegant, but:
The lowest possible for x1 is 5, forcing (x2,x3)=(10,10).
Fix x1=6 then it's either (x2,x3)=(9,10) or (10,9)
Fix x2=7 then it's either (x2,x3)=(8,10) or (10,8) or (9,9)
As you can see there really aren't that many options if you continue in this manner.
A generating function solution requires the least thought, in my opinion.
Suppose, more generally, we want to count the integer solutions to
where ;
let's say the number of solutions is .
Then is the coefficient of in
.
From here on it's just a matter of manipulating series:
Picking out the coefficient of from this expression, we can see it is
.
I think a solution via inclusion/exclusion is also possible (I haven't worked it out), but I think it will lead to the same expression for the answer. Only you will have to think harder. :-P