Enumerating sums of integers So I found a question which seems to be really easy to answer but after thinking a lot about it, well, I've have to say, I came up with nothing.
So here's the question:

Assume the sum of $a+b+c+d+e+f$ that each of the numbers can be $0,3,4,5$, how many different summations can exist?

The only point I came up with is that the integer $0$ doesn't change the summation therefore different permutations of a $0$ count as one as I said the reason above.
And sorry if the Tags are irrelevant and my Math language is crappy I'm a newbie!
Please feel free to edit my tags I really do not know which one to add.
Any help would be highly appreciated in advance.

Denote the set of all possible sums by $S$. Clearly $S\subset\{0,1,2,3,\cdots,30\}$.
Then take care of each element in $\{0,1,\cdots,30\}$.
$30=5+5+5+5+5+5$
$29=5+5+5+5+5+4$
$28=5+5+5+5+5+3$
$27=5+5+5+5+4+3$
$26=5+5+5+5+3+3$
Replacing one 5 by 0, we can obtain $25,24,23,22,21$. Repeat this process for 4 times until we get
$10=0+0+0+0+5+5$
$9=0+0+0+0+5+4$
$8=0+0+0+0+5+3$
$7=0+0+0+0+4+3$
$6=0+0+0+0+3+3$.
Then we see $S=\{0,3,4,5\}\cup\{6,7,\cdots,30\}$.
