How many non negative integer solutions are there for the equation $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 36$ with restrictions? So I had a few enquiries with this question:
Consider the non negative integer solutions for the equation $$x_1 + x_2 + x_3 + x_4 + x_5 = 36$$
a) How many distinct solutions are there?
b) How many distinct solutions are there if $x_1 \geq 12$?
c) How many distinct solutions are there if $x_1 < 18$?
d) How many distinct solutions are there if $x_1 < 18$ and $x_2 < 6$?
I was wondering if distinct is the same as unique? as in there cannot be solutions that have the same numbers but rearranged?
Also, how would one approach this question. Is there a principle that would be best applied to this question?
 A: You can approach the solution using a simple visual.
Assuming $5$ variables, consider a line of 36 stars,  $*$, and $4$ bars, $|$.
For example ******|*****|*****|********|************
Consider that the $4$ vertical bars divide the $36$ stars into $5$ "bins", the above row represents the particular solution $(6, 5, 5, 8, 12)$
Now all you need to do is count the arrangements of $40$ objects, with $4$ bars all identical to each other, and the remaining $36$ stars all identical to each other.
For the second part, put $12$ stars aside, repeat this calculation with the reduced set of stars.  For each of the solutions you get, picture then dumping the $12$ removed stars into Bin #$1$. What do you have?
For the last two problems, consider that you have just found how many solutions there are, and how many have $x_1$ equal to or greater than $12$.  So the difference must be the solutions that have $x_1$ less than $12$. Does that help with the last two parts?
Googling on "Stars and Bars" will give a lot more information on this technique.
