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Math Help - Blackjack hands

  1. #1
    Member oldguynewstudent's Avatar
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    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?
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  2. #2
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    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 A\heartsuit.

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


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

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


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

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