number of non-negative integer solutions of $ax+by+cz=k$ How to find the number of non-negative integer solutions of $2x+3y+z=21$ ? 
Please Help , thanks in advance 

We can look at the number of $(x,y)$ pairs such that $2x+3y\le 21$.
We have:
$$(0,0),(0,1),\dots,(0,7)$$
$$(1,0),\dots,(1,6)$$
$$(2,0),\dots,(2,5)$$
$$\vdots$$
$$(10,0)$$
The maximum $y$ takes the value $\lfloor \frac{21-2x}{3} \rfloor$, i.e.
$$\begin{array} {c|c}
x&\lfloor \frac{21-2x}{3} \rfloor\\
\hline
0&7\\
1&6\\
2&5\\
3&5\\
4&4\\
5&3\\
6&3\\
7&2\\
8&1\\
9&1\\
10&0\\
\hline
\sum&37
\end{array}
$$
The lists are zero-based, so the final answer is $37+11=48$.
