Find the number of integer solutions for $x_1+x_2+x_3\le15$

Question

For equation $x_1+x_2+x_3\le15$

Find the number of positive (including 0) integer solutions on conditions:

$x_1\ge4$

$x_3 \in \{2,5,7\}$

I have tried using inclusion-exclusion

Any ideas?

Answer 1

You neither need nor want to use the principle of inclusion and exclusion. Solve for $x_1+x_2+x_3+x_4 = 11$. The solutions to this are in $1$-$1$ correspondence with the solutions to your system with $x_3$ unconstrained. Now just count how many different solutions for each value of $x_3$.

For $x_3=2$: How many solutions to $x_1+x_2+x_4=9$? $\binom{11}{2}=55$

For $x_3=5$: How many solutions to $x_1+x_2+x_4=6$? $\binom{8}{2}=28$

For $x_3=7$: How many solutions to $x_1+x_2+x_4=4$? $\binom{6}{2}=15$

Use stars and bars to answer these three questions as above, then add the answers to see that there are $98$ possible solutions.