You own 6 dogs, 4 cats and 5 rabbits. In how many ways can you pet them if you have to pet 4 dogs, 3 cats and 2 rabbits (You are allowed to pet the same animal multiple times).
I tried doing 6^4 * 4^3 * 5^2 * 9!, but apparently the number is too large because it doesn't account for the repetitions.
This is a difficult question to answer, because we're missing some information. Does order matter or not? If it does, then we're looking at how to permute all 9 pettings in different ways. Then, we start by finding the ways to permute the pettings if we only look at species rather than individual animal. There are $\frac{9!}{4!3!2!}$ ways to perform this permutation. Then, we have to multiply by the number of ways to choose animals within the species, which you correctly calculated as $6^4 * 4^3 * 5^2$.