This Week's Work : Week 23 - for Inquisitive Young Mathletes

Sticks and Stones Method to answer question such as :
How many ordered triples (x, y, z) of natural numbers satisfy x + y + z = 5 ?

x + y + z = 5  and x, y, z are natural numbers (given)

You can look at this as 5 stones. O O O O O  if you place the two sticks (3 numbers so 3 - 1 = 2 sticks
will help break it into three groups)

You can get all the x, y and z combinations.
O | OO | OO means x = 1 , y = 2 and z = 2
O O | OO | O means z = 2 , y = 2 and z = 1
There are 4 spaces and you are placing 2 sticks in the 4 spaces so 4C2 or 6 ways.

If x + y + z = 5 and all three could be "0", then you do  (5 + 3 (x, y and z) -1) choose 2 (3 numbers so 3-1)
or 7C2
so O O O |O O | O O O means 2, 1, 2
O|O O O | OOOO means O, 2 and 3  (in this case, "O" means the number o, "OO" means the number 1
and "OOO" means the number 2. Make sure you understand this !!)
7 because there are 7 spaces in between those stones.

Another way is to still use OOOOO ||. There are 7!/5!*2! or 7C2 to arrange the stones and sticks.
  |  |OOOOO means 0 + 0 + 5 (nothing on the left of the divider, nothing in between and 5 on the right
of the divider.
  |OO|OOO means 0 + 2 + 3
Thus, the answer is still 7C2

More sticks and stones practices from AoPs book excerpt
More sticks and stones questions from my blog