From a deck of 52 cards, extract 10. In how many combinations do you get at least one ace?

Question

If from a deck of 52 cards, I extract 10. In how many combinations do you get at least one ace?

I have come up with two possible answers, but I don't know which one is the right one and why.

So one is $$4{51 \choose 9}$$ the reasoning is that first I extract an ace from the four that there are, and then I have ${51 \choose 9}$ combinations for the other 9 cards.

The second is $${52 \choose 10} - {48 \choose 10}$$ reasoning that there are ${52 \choose 10}$ total combinations of ten cards and I subtract ${48 \choose 10}$ combinations without any aces.

So from other questions it seems that the first one is the correct answer, but why would the second one be wrong?

Thanks for the answers.

Answer 1

To see why your first answer is not correct, assume you have a deck of 3 with 2 aces and you'll draw 2 cards with at least one ace. Call the cards $A_1,A_2,B$

Based on your logic, you compute $2 {2 \choose 1}=4$ combinations. However, clearly you have only $3$. This is due to double counting $\{A_1,A_2\}$

Answer 2

The first approach needs to be modified to ${4\choose1}{48\choose 9}+{4\choose2}{48\choose 8}+{4\choose 3}{48\choose 7}+{4\choose 4}{48\choose 6}$, for instance.

The second is correct.