Determine the number of integer solutions of X1 +X2 +X3 +X4 +X5 < 40
where
a) Xi > 0, 1 <= i <= 5
b) Xi >= -3, 1 <= i <= 4 and X5 => 3
Hello,
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 $\displaystyle X_6$ a positive integer $\displaystyle (>0)$ and $\displaystyle Y_6$ a nonnegative integer $\displaystyle (\geq 0)$
For question 1), write the inequality as $\displaystyle X_1+X_2+X_3+X_4+X_5+X_6=40$ (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 $\displaystyle X_1+X_2+X_3+X_4+X_5+Y_6=39$
Let $\displaystyle Y_i=X_i+3 ~,~ i\in\{1,2,3,4\}$ and $\displaystyle Y_5=X_5-3$
Substitute the $\displaystyle X_i$ by the $\displaystyle Y_i$ in the inequality to get :
$\displaystyle Y_1+Y_2+Y_3+Y_4+Y_5+Y_6=39$, where $\displaystyle Y_i\geq 0 ~,~ i\in\{1,2,3,4,5,6\}$
and you can apply theorem 2.