TITLE: Find the number of integer non-negative solutions to $3 x_1+5 x_2+x_3+x_4=10.$
QUESTION [7 upvotes]: Find the number of integer non-negative solutions to
$$3 x_1+5 x_2+x_3+x_4=10.$$

My proposed solution: First find the possible values for the pairs $(x_1,x_2)$ which are allowed: $(0,0),(0,1),(0,2),(1,0),(1,1),(2,0),(3,0)$, and count solutions for $x_3, x_4$ in each case. Counting for each case is, respectively: 11,6,1,8,3,5,2. They add to a total of 36 solutions.
My problems: Given solution in a book is 30 and I would like to know whether my solution is indeed wrong. Is there a more elegant way to solve this problem and similar problems? I believe my approach is kind of brute force.

REPLY [5 votes]: your result is true the number of solutions are $36$ for sure.
all solutions are listed here:
$ 1 :    x_1=0, x_2=0, x_3=0, x_4=10$
$ 2 :    x_1=0, x_2=0, x_3=1, x_4=9$
$ 3 :    x_1=0, x_2=0, x_3=2, x_4=8$
$ 4 :    x_1=0, x_2=0, x_3=3, x_4=7$
$ 5 :    x_1=0, x_2=0, x_3=4, x_4=6$
$ 6 :    x_1=0, x_2=0, x_3=5, x_4=5$
$ 7 :    x_1=0, x_2=0, x_3=6, x_4=4$
$ 8 :    x_1=0, x_2=0, x_3=7, x_4=3$
$ 9 :    x_1=0, x_2=0, x_3=8, x_4=2$
$ 10 :    x_1=0, x_2=0, x_3=9, x_4=1$
$ 11 :    x_1=0, x_2=0, x_3=10, x_4=0$
$ 12 :    x_1=0, x_2=1, x_3=0, x_4=5$
$ 13 :    x_1=0, x_2=1, x_3=1, x_4=4$
$ 14 :    x_1=0, x_2=1, x_3=2, x_4=3$
$ 15 :    x_1=0, x_2=1, x_3=3, x_4=2$
$ 16 :    x_1=0, x_2=1, x_3=4, x_4=1$
$ 17 :    x_1=0, x_2=1, x_3=5, x_4=0$
$ 18 :    x_1=0, x_2=2, x_3=0, x_4=0$
$ 19 :    x_1=1, x_2=0, x_3=0, x_4=7$
$ 20 :    x_1=1, x_2=0, x_3=1, x_4=6$
$ 21 :    x_1=1, x_2=0, x_3=2, x_4=5$
$ 22 :    x_1=1, x_2=0, x_3=3, x_4=4$
$ 23 :    x_1=1, x_2=0, x_3=4, x_4=3$
$ 24 :    x_1=1, x_2=0, x_3=5, x_4=2$
$ 25 :    x_1=1, x_2=0, x_3=6, x_4=1$
$ 26 :    x_1=1, x_2=0, x_3=7, x_4=0$
$ 27 :    x_1=1, x_2=1, x_3=0, x_4=2$
$ 28 :    x_1=1, x_2=1, x_3=1, x_4=1$
$ 29 :    x_1=1, x_2=1, x_3=2, x_4=0$
$ 30 :    x_1=2, x_2=0, x_3=0, x_4=4$
$ 31 :    x_1=2, x_2=0, x_3=1, x_4=3$
$ 32 :    x_1=2, x_2=0, x_3=2, x_4=2$
$ 33 :    x_1=2, x_2=0, x_3=3, x_4=1$
$ 34 :    x_1=2, x_2=0, x_3=4, x_4=0$
$ 35 :    x_1=3, x_2=0, x_3=0, x_4=1$
$ 36 :    x_1=3, x_2=0, x_3=1, x_4=0$