How many ways to select 8 animals among goats, sheep, cows, horses and dogs? The solution given considered it identical to finding the number of integral solutions to $$a+b+c+d=8$$
I know how to solve this. It's a straightforward stars and bars question. However, I approached the question in the following way - Each animal can be chosen in $4$ ways so the answer is $8^4$. 
Although I understand the solution given, I don't understand these two approaches don't count the same entity. Can someone tell the mistake in my reasoning ? 
 A: To begin with, there's 5 types of animals.  But let's pretend there's no dogs.


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*$4^8$ counts length-8 sequences with elements chosen from {goat,sheep,cow,horse}.  So, e.g.,
goat goat goat horse horse horse goat goat

is one sequence and
goat goat goat goat goat horse horse horse

is another.  There are eight "slots", and for each slot, we can choose one of four animals.

*$8^4$ seems to be the above case with the numbers ($8$ and $4$) misplaced.
The first bullet point overcounts because we're counting selections of the same animals multiple times.


*

*The stars and bars method works fine, e.g. $$\star \star | \ | \star \star \star \star \star | \star$$ corresponds to "2 goats, 0 sheep; 5 cows; 1 horse", and we don't account for whether or not we choose goats before cows, etc.

A: Counting 4 choices for each of the 8 slots double counts permutations. We don't care about the order in which the animals are chosen. So if you pick 3 goats, then 4 cows, and then another goat, that is the same as picking 4 goats and 4 cows. Your method, however, would count these as two distinct scenarios. Also, I believe you need five variables, since there are five types of animals according to the title.
