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Math Help - Poker Question

  1. #1
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    Poker Question

    Hi all

    The question is:

    Given a 52 card deck, how many hands of 5 cards have

    a) exactly 3 aces?

    My guess is 4 x 3 x 2 x 48 x 47. Although this is not correct according to my text book.

    b) at least 3 aces?

    My guess is, having 3 aces plus having 4 aces = (4 x 3 x 2 x 48 x 47) + (4 x 3 x 2 x 1 x 48) . This is also not correct
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  2. #2
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    Quote Originally Posted by Bushy View Post
    Hi all

    The question is:

    Given a 52 card deck, how many hands of 5 cards have

    a) exactly 3 aces?

    My guess is 4 x 3 x 2 x 48 x 47. Although this is not correct according to my text book.

    b) at least 3 aces?

    My guess is, having 3 aces plus having 4 aces = (4 x 3 x 2 x 48 x 47) + (4 x 3 x 2 x 1 x 48) . This is also not correct
    For part (a) you can modify slightly to get the correct answer.

    (4*3*2)/3! * (48*47)/2!

    This is also C(4,3) * C(48,2) where C(n,k) is binomial coefficient.

    4*3*2 means order matters; 4*3*2/3! means order does not matter.

    For part (b) you can do as you suggested, but you will need to modify your expression for 4 aces.
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  3. #3
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    Hello, Bushy!

    Are you familiar with Combinations?


    Given a 52-card deck, how many hands of 5 cards have:

    (a) exactly 3 Aces?

    There are 4 Aces and 48 Others.

    We want 3 Aces and 2 Others.

    . . There are: . _4C_3 = 4 ways to get 3 Aces.

    . . There are: . _{48}C_2 = 1128 ways to get 1 Other.

    Therefore: . 4\cdot1128 \:=\:4512 hands with exactly 3 Aces.




    (b) at least 3 Aces?

    "At least 3 Aces" means: . \begin{Bmatrix}\text{3 Aces, 2 Others} \\ \text{or} \\ \text{4 Aces, 1 Other} \end{Bmatrix}


    3 Aces, 2 Others: from part (a), 4512 ways.


    4 Aces, 1 Other:
    . . . There is: . _4C_4 \:=\:1 way to get 4 Aces.
    . . There are: . _{48}C_1 \:=\: 48 ways to get 1 Other.

    Hence, there are: . 1\cdot48\:=\:48 ways to get 4 Aces, 1 Other.


    Therefore: . 4512 + 48 \:=\:4560 ways to get at least 3 Aces.
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