I wish to improve on my combinatorial reasoning skills and my step-father gave me this problem that has left me quite confused. It seems to me that because the balls are colored similar to the boxes, that you have to factor in the ball and box colors. I do not wish to have the answers just "given" I wish to develop the right thinking that could help me solve the problem. Thank you.
There are 3 boxes (red, green, blue) and 7 distinguishable balls (two red balls, three blue balls, and two green balls).
a. Find the number of ways to put the balls into the boxes with no restrictions.
For the first part, I believe the solution is just $3^7$ since we were told there are no restrictions. I take this to mean that we can treat all 7 balls as distinguishable, so we don't care that there are sub-groupings of ball colors. Is this valid?
b. Find the number of ways to put the balls into the boxes so that each ball is not in its color's box (seems similar to the idea of derangements)
1 red box and 5 non-red balls, 1 blue box and 4 non-blue balls 1 green box and 5 non-green balls
c. Find the number of ways to place the balls into the boxes so that each box is non-empty (definitely a Stirling number of the second kind problem)
d. Find the number of ways to place the balls so that all balls of the same color are together, each ball is not in its colors box , and each box is not empty. (combines answers)