TITLE: How many integer solutions are there to the inequality $y_1 + y_2 + y_3 + y_4 < 184$
QUESTION [5 upvotes]: $y_1 + y_2 + y_3 + y_4 < 184$
$y_1>0$ 
$0< y_2\leq10$
$0\leq y_3\leq17$ 
$0\leq y_4<19$
Don't really know what to do with the inequality. Can I simply change it to $y_1 + y_2 + y_3 + y_4  + y_5 = 183$? If so, will $y_5\geq 0$?
Edit: So this is what I got, it may or may not be correct:
$y_1 + y_2 + y_3 + y_4  + y_5 = 181$
$y_1\geq0, 0\leq y_2\leq9, 0\leq y_3\leq17, 0\leq y_4<19$
(181+4)C4 - (181+4-10)C4 - (181+4-18)C4 - (181+4-19)C4 + (181-19-18-10+4)C4

REPLY [2 votes]: For given values of $y_2$, $y_3$, and $y_4$, there are $183-y_2-y_3-y_4$ valid choices for $y_1$, so the number is $$\sum_{y_2=1}^{10} \sum_{y_3=0}^{17} \sum_{y_4=0}^{19} (183-y_2-y_3-y_4).$$
We simplify this as follows (using triangular numbers):
\begin{align*}
& \phantom{={}} \sum_{y_2=1}^{10} \sum_{y_3=0}^{17} \sum_{y_4=0}^{19} (183-y_2-y_3-y_4) \\
& = \sum_{y_2=1}^{10} \sum_{y_3=0}^{17} \sum_{y_4=0}^{19} 183 - \sum_{y_2=1}^{10} \sum_{y_3=0}^{17} \sum_{y_4=0}^{19} y_2 - \sum_{y_2=1}^{10} \sum_{y_3=0}^{17} \sum_{y_4=0}^{19} y_3 - \sum_{y_2=1}^{10} \sum_{y_3=0}^{17} \sum_{y_4=0}^{19} y_4 \\
& = 10 \times 18 \times 20 \times 183-18 \times 20 \times \tfrac{10 \times 11}{2}-10 \times 20 \times \tfrac{17 \times 18}{2}-10 \times 18 \times \tfrac{19 \times 20}{2} \\
& = 658800 - 19800 - 30600 - 34200 \\
& = 574200.
\end{align*}
We can verify this by generating all of them using the GAP code:
count:=0;;
for y2 in [1..10] do
  for y3 in [0..17] do
    for y4 in [0..19] do 
      for y1 in [1..183-y2-y3-y4] do
        count:=count+1;
        Print(count," ",[y1,y2,y3,y4],"\n");
      od;
    od;
  od;
od;

whose output ends with:
574197 [ 134, 10, 17, 19 ]
574198 [ 135, 10, 17, 19 ]
574199 [ 136, 10, 17, 19 ]
574200 [ 137, 10, 17, 19 ]

NB. This answer is almost identical to my other answer here.