How many solutions does the equation $x+y+z=17$ have where $x,y,z$ are non negative integers?

Question

I'm not able to understand the solution to this problem. It says selecting $17$ items from a set of $(x,y,z)$ Final answer: $$\binom{17+3-1}{17}=\binom{19}{17}$$

Answer 1

This is a classical Stars and Bars problem. Using the explanation in the link it's not too dificult to conlcude that the wanted answer is:

$$\binom{17+3-1}{17} = \binom{19}{17}$$

Answer 2

First write 17 number of 1's in a row. Now we want to arrange 17 number of 1's into three groups containing zero or more 1's. So, we put two stars in between these 1's which will give three group of 1's. The arrangement may look something like this: $$111\cdots1\ast 11\cdots 1\ast 111\cdots 1$$ or $$\ast \ast 111\cdots 1.$$ In the second arrangement, there is zero 1's before first star and second star, and all 1's are after second star. So, we have the grouping 0, 0, 17 which gives a solution. Thus we see that we need to place these two stars at two places to form three groups of 1's. Since total number of objects are 17 + 2 : 17 one's and 2 stars. Hence we can place 2 stars anywhere among these 19 places in $19\choose2$ ways, which is the number of solutions. More generally, by the similar argument, we can show that the number of nonnegative solutions of $x_1+x_2+ \cdots +x_k =h$ is $k+h-1 \choose h-1$.