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Thread: how many different ways to hit 21

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    how many different ways to hit 21

    this may seem like an odd or stupid question, but how many different ways can you get 21 with numbers 2 through 9 in a three digit combo?
    example:
    9+9+3
    9+8+4
    9+7+5
    9+6+6
    8+8+5
    8+7+6
    7+7+7
    is that it?

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    If you want 3 card combinations:
    9,9,3: 3 combinations (993, 939, and 993)
    9,8,4: 6 combinations (984, 948, 894, 849, 489, and 498)
    9,7,5: 6 combinations
    9,6,6: 3 combinations
    8,8,5: 3 combinations
    8,7,6: 6 combinations
    7,7,7: 1 combination
    Total number of combinations is 28.

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    ayjacks,

    "is that it?"

    NO !

    There are many multi-card combinations.

    Here is a list of them all:

    I found this in the stacks of the SPECIAL collection at the University of Nevada at Las Vegas.

    EDIT: OOPS! This is for 16, not 21.

    Attached Images Attached Images

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    Quote Originally Posted by Tthree View Post
    If you want 3 card combinations:
    9,9,3: 3 combinations (993, 939, and 993)
    9,8,4: 6 combinations (984, 948, 894, 849, 489, and 498)
    9,7,5: 6 combinations
    9,6,6: 3 combinations
    8,8,5: 3 combinations
    8,7,6: 6 combinations
    7,7,7: 1 combination
    Total number of combinations is 28.
    Better yet, how many different combinations can make 21 using every possible combination of all 4 suits or less (2 cards is the least), with 52 cards to make 21?

    28 seems inconclusive T3, as there are 4 suits that hold the value of a 7 pip card in a single deck. So there should be 4 different combinations of 21 using only the 7's in a single deck as I have listed below. Not 1 combination.

    The OP didn't state the number of decks in use in his original question. However, as the number of decks increases to 2 decks, the number of different combinations totaling 21 increases accounting for only the 7. If there are 3 decks in play, then there is the possible combination of 3 similar 7's being dealt such as 7s, 7s, 7s. In 3 decks what is the frequency that any player would be dealt 3-of-a-kind suited on the first 3 cards of any particular hand before the cut card with one player at the table, even if the final hit was a possible suited bust card?

    7h, 7d, 7s red-red-black
    7h, 7s, 7c red-black-black
    7d, 7c, 7s red-black-black
    7d, 7h, 7c red-red-black
    Last edited by Blitzkrieg; 01-05-2016 at 03:54 AM.

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    The OP specifically asked about 3 card combinations.

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    Quote Originally Posted by Gronbog View Post
    The OP specifically asked about 3 card combinations.
    Noted Gronbog. I was merely pointing out to T3 that I see 4 different 3 card combinations for 7's, not 1 in a single deck. To which T3 must surely concur.

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    Quote Originally Posted by Blitzkrieg View Post
    Noted Gronbog. I was merely pointing out to T3 that I see 4 different 3 card combinations for 7's, not 1 in a single deck. To which T3 must surely concur.
    For most things card suits are irrelevant in the casino. There are exceptions. Unless otherwise stated I assume ranks are suitless when asked a question. But if you want to consider suit there are 48 different ways to get 777.
    7c, 7c, 7c
    7c, 7c, 7d
    7c, 7c, 7h
    7c, 7c, 7s
    7c, 7d, 7c
    7c, 7d, 7d
    7c, 7d, 7h
    7c, 7d, 7s
    7c, 7h, 7c
    7c, 7h, 7d
    7c, 7h, 7h
    7c, 7h, 7s
    7c, 7s, 7c
    7c, 7s, 7d
    7c, 7s, 7h
    7c, 7s, 7s

    7d, 7c, 7c
    7d, 7c, 7d
    7d, 7c, 7h
    7d, 7c, 7s
    7d, 7d, 7c
    7d, 7d, 7d
    7d, 7d, 7h
    7d, 7d, 7s
    7d, 7h, 7c
    7d, 7h, 7d
    7d, 7h, 7h
    7d, 7h, 7s
    7d, 7s, 7c
    7d, 7s, 7d
    7d, 7s, 7h
    7d, 7s, 7s

    7h, 7c, 7c
    7h, 7c, 7d
    7h, 7c, 7h
    7h, 7c, 7s
    7h, 7d, 7c
    7h, 7d, 7d
    7h, 7d, 7h
    7h, 7d, 7s
    7h, 7h, 7c
    7h, 7h, 7d
    7h, 7h, 7h
    7h, 7h, 7s
    7h, 7s, 7c
    7h, 7s, 7d
    7h, 7s, 7h
    7h, 7s, 7s

    7s, 7c, 7c
    7s, 7c, 7d
    7s, 7c, 7h
    7s, 7c, 7s
    7s, 7d, 7c
    7s, 7d, 7d
    7s, 7d, 7h
    7s, 7d, 7s
    7s, 7h, 7c
    7s, 7h, 7d
    7s, 7h, 7h
    7s, 7h, 7s
    7s, 7s, 7c
    7s, 7s, 7d
    7s, 7s, 7h
    7s, 7s, 7s

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    Quote Originally Posted by Tthree View Post
    For most things card suits are irrelevant in the casino. There are exceptions. Unless otherwise stated I assume ranks are suitless when asked a question. But if you want to consider suit there are 48 different ways to get 777.
    7c, 7c, 7c
    7c, 7c, 7d
    7c, 7c, 7h
    7c, 7c, 7s
    7c, 7d, 7c
    7c, 7d, 7d
    7c, 7d, 7h
    7c, 7d, 7s
    7c, 7h, 7c
    7c, 7h, 7d
    7c, 7h, 7h
    7c, 7h, 7s
    7c, 7s, 7c
    7c, 7s, 7d
    7c, 7s, 7h
    7c, 7s, 7s

    7d, 7c, 7c
    7d, 7c, 7d
    7d, 7c, 7h
    7d, 7c, 7s
    7d, 7d, 7c
    7d, 7d, 7d
    7d, 7d, 7h
    7d, 7d, 7s
    7d, 7h, 7c
    7d, 7h, 7d
    7d, 7h, 7h
    7d, 7h, 7s
    7d, 7s, 7c
    7d, 7s, 7d
    7d, 7s, 7h
    7d, 7s, 7s

    7h, 7c, 7c
    7h, 7c, 7d
    7h, 7c, 7h
    7h, 7c, 7s
    7h, 7d, 7c
    7h, 7d, 7d
    7h, 7d, 7h
    7h, 7d, 7s
    7h, 7h, 7c
    7h, 7h, 7d
    7h, 7h, 7h
    7h, 7h, 7s
    7h, 7s, 7c
    7h, 7s, 7d
    7h, 7s, 7h
    7h, 7s, 7s

    7s, 7c, 7c
    7s, 7c, 7d
    7s, 7c, 7h
    7s, 7c, 7s
    7s, 7d, 7c
    7s, 7d, 7d
    7s, 7d, 7h
    7s, 7d, 7s
    7s, 7h, 7c
    7s, 7h, 7d
    7s, 7h, 7h
    7s, 7h, 7s
    7s, 7s, 7c
    7s, 7s, 7d
    7s, 7s, 7h
    7s, 7s, 7s
    This is the type of response that I was expecting to see from you when replying to the OP. A thorough approach at solving the problem in part to show the OP how there are many possibilities for what he is asking. Card suits do have relevance in the casino as you well know.

    I am delighted that you were up to the task of writing out 3 decks of 7's. I noticed that you wrote it out in minor suit to major suit format in the left, middle, and far right column (cdhs) in a downward fashion and cycled through all the combinations as I would have for ease. According to what you typed out for 3 decks which I looked over to ensure it was all there, there are 64 ways instead of 48 or did I miss something?
    Last edited by Blitzkrieg; 01-07-2016 at 01:07 AM.

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