Incorrect combinatorics reasoning There are $5$ cows, $8$ roosters, and $10$ pigs on a farm. The farmer wants to pick $4$ animals, and at least $1$ needs to be a cow. He asked his (alleged) prodigious son Smarty how many ways it can be done.
Smarty says "This is so easy." You pick a cow in ${5\choose 1}$ ways and pick $3$ of the remaining $22$ animals in ${22\choose 3}$ ways. So he says the answer is ${5\choose 1}{22\choose 3}$. But the farmer shakes his head in disgust and says this is incorrect.

Why is smarty wrong? I would count it in the same way. Can someone please clarify?
 A: Smarty is wrong because more than one cow may be picked. The choice in which they picked Daisy first and then Clover was picked as one of the 22 other animals is wrongly considered different from the pick where Clover was picked first and then Daisy was one of the 22 other animals.
A: Add $$\dbinom{5}{1}\dbinom{18}{3}+\dbinom{5}{2}\dbinom{18}{2}+\dbinom{5}{3}\dbinom{18}{1}+\dbinom{5}{4}\dbinom{18}{0}$$ because it says at least one must be a cow (so you can have two, three, or four cows picked). The other way to do it: $$\dbinom{23}{4} - \dbinom{18}{4}$$
A: Let's say that $C_1,C_2,\dots,C_5$ are the cows, and $R_1,\dots,R_8$ are the roosters and $P_1,\dots,P_{10}$ are the pigs. Here are two ways of carrying out the procedure which results in $\binom{5}1\cdot \binom{22}3$. 


*

*From the cows, choose $C_1$. Then from the other animals, choose $C_3, R_1, P_2$.

*From the cows, choose $C_3$. Then from the other animals, choose $C_1,R_1,P_2$.
Note that these result in the same set of animals, $\{C_1,C_3,R_1,P_2\}$. This set has been overcounted, so $\binom51\cdot\binom{22}3$ is too large. 
Here is another way to think about it; by choosing one of the cows, then choosing the other animals, the first chosen cow becomes "special," as it was chosen in a different step of the process then the other cows. Smarty's procedure is therefore not counting subsets of four animals with at least one cow, but is actually counting subsets of four animals with at least one cow and where one of the cows is special. The latter is more specific, and can done in more ways.
