Suppose that we have 3 red balls, 6 blue balls, and 7 green balls. Determine the number of ways to put 8 balls into one box I can tackle this problem by first listing out the # of red balls: 0,1,2,3. But this seems brute force to me and I don't know what to do if the number and the variety goes up, Can someone give me a hint on a more intuitive approach?((balls with the same color are considered identical))

A way to use stars and bars here is to imagine that the balls are identical, and that there are $3$ magical boxes that change the color of the ball put in to red, blue or green.
We also note that at most one box can violate the constraints, which simplifies working greatly.
Using stars and bars, and excluding forbidden combos by pre-placing $4,7$ and $8$ balls respectively in the red, blue and green box,
number of ways $= \dbinom{10}2 - \dbinom62 - \dbinom32 - \dbinom22 = 26$ 
