Determine the number of integer solutions of X1 +X2 +X3 +X4 +X5 < 40
a) Xi > 0, 1 <= i <= 5
b) Xi >= -3, 1 <= i <= 4 and X5 => 3
The method I'll present may look a tad weird, and experimental, so don't hesitate if you have any objection lol.
You basically need the two theorems in there : Stars and bars (probability) - Wikipedia, the free encyclopedia
First let a positive integer and a nonnegative integer
For question 1), write the inequality as (think on your own to know why it's correct) and apply theorem 1 (because we have strict inequalities for the unknowns).
For question 2), we have
Substitute the by the in the inequality to get :
and you can apply theorem 2.