We have 40 cards. A, B, C extract resp. 5,4,3 cards. How many extractions can we make in which A and B don't extract swords while C extract at least 2? Solution: $C_{30,5}\cdot C_{25,4}\cdot C_{10,2}\cdot C_{29,1}$.
I wrote: $C_{30,5}\cdot C_{25,4}\cdot C_{10,2}\cdot C_{21,1}+C_{30,5}\cdot C_{25,4}\cdot C_{10,3}$. Why is my solution wrong?
The reference to a $40$-card deck with swords as a suit means this is a Spanish deck with $10$ each of swords, cups, coins and clubs, not the "standard" (French) $52$-card deck.
Your solution is the correct one, distinguishing between two-sword and three-sword cases. The given solution overcounts those cases where C's hand is all swords: it counts picking ($\{2,3\}$ and then $4$ of swords) and ($\{4,3\}$ and then $2$ of swords) as distinct, when they are not.