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
The face-down card is the : 1 way
The face-up is an Other: 39 ways.
. . Case (1) has: . hands.
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.