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

. . (2) The face-down card is the

Case (1)

The face-down card is the : 1 way

The face-up is an Other: 39 ways.

. . Case (1) has: . hands.

Case (2)

The face-down card is the 3 ways.

The face-up card is an Other: 38 ways.

. . Case (2) has: . hands.

Therefore, there are: . hands.