# In how many ways can $12$ balls be distributed to $4$ distinguishable bags if the balls are distinguishable and each bag gets at least 3 balls?

In how many ways can $12$ balls be distributed to $4$ distinguishable bags if the balls are distinguishable and each bag gets at least 3 balls?

If each bag gets 3 balls then there are 0 balls left. But that will be true if the balls are identical. I am not sure how to do if the balls are nonidentical.

• What do you think? (P.S., this is not a 'do my homework for free' service). – barak manos May 1 '16 at 20:10
• I think just one 1 way but that would be true of the balls are identical. – user336166 May 1 '16 at 20:11
• And why do you think that? Can you show this one way and explain why there aren't any others? (within the question please). – barak manos May 1 '16 at 20:12
• what do you mean? in the question that balls are not identical. – user336166 May 1 '16 at 20:14
• I have made the title more descriptive. Please edit your question to indicate what you have attempted and where you are stuck so that you receive responses that address the specific difficulties you are encountering. – N. F. Taussig May 1 '16 at 20:18

Since both the balls and the bags are distinguishable, what matters here is which balls are placed in which bag. Line up the bags in some order (for instance, by size or color). Choose which three of the $12$ balls go in the first bag, which three of the nine remaining balls go in the second bag, which three of the six remaining balls go in the third bag, and place the rest in the fourth bag.
$$\binom{12}{3}\binom{9}{3}\binom{6}{3}\binom{3}{3}$$