1. ## Blackjack hands

How many hands are there in which the face-down card is an ace and the face-up card is not a heart?

There are 4 ways to choose an ace times 39 ways to choose not a heart, then subtract out the combinations of two ace of spades, two ace of diamonds and two ace of clubs. This gives 4*39 - 3 = 153. Is this the correct answer?

2. Hello, oldguynewstudent!

How many hands are there in which the face-down card is an Ace
and the face-up card is not a heart?

There are 4 ways to choose an ace times 39 ways to choose not a heart,
then subtract out the combinations of two ace of spades, two ace of diamonds
and two ace of clubs.

This gives 4*39 - 3 = 153.
Is this the correct answer? . Yes!

I solved it like this . . .

We have 13 Hearts and 39 Others.

There are two cases to consider:

. . (1) The face-down card is the $\displaystyle A\heartsuit.$

. . (2) The face-down card is the $\displaystyle A\spadesuit,\,A\diamondsuit, \text{ or }A\clubsuit.$

Case (1)
The face-down card is the $\displaystyle A\heartsuit$: 1 way
The face-up is an Other: 39 ways.
. . Case (1) has: .$\displaystyle 1\cdot39 \:=\:39$ hands.

Case (2)
The face-down card is the $\displaystyle A\spadesuit,\,A\diamondsuit,\text{ or } A\clubsuit:$ 3 ways.
The face-up card is an Other: 38 ways.
. . Case (2) has: .$\displaystyle 3\cdot38 \:=\:114$ hands.

Therefore, there are: .$\displaystyle 39 + 114 \:=\:153$ hands.