How many integer solutions (x1; x2; x3) are there to the equation :-
x1 + x2 + x3 = 25
with all xi satisfying 0 < xi < 10?
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.
Suppose, more generally, we want to count the integer solutions to
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