I have a bag full of: 7 green rocks, 12 yellow rocks, and 15 red rocks. How many ways are there to reach in and grab 4 rocks?
Is the answer 37C34 (37=7+12+15+4-1) or 6C3 (6=3+4-1)...or something else? By "C", I mean combination. This is a multiset problem as repeats are allowed and order doesn't matter.
Assuming rocks of the same color are indistinguishable, the number of ways of grabbing four rocks from the bag is equal to the number of solutions of the equation $$g + y + r = 4$$ in the nonnegative integers, where $g$ is the number of green rocks, $y$ is the number of yellow rocks, and $r$ is the number of red rocks that are selected. The number of solutions of the equation $g + y + r = 4$ in the nonnegative integers is the number of ways two addition signs can be placed in a row of four ones, which is $$\binom{4 + 2}{2} = \binom{6}{2}$$