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Math Help - Card game: number of combinations

  1. #1
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    Card game: number of combinations

    In a card game an Ace = 4 points, king=3 points, queen=2 points and jack=1 point.
    Now we have only 16 cards that have points to deal with(4A,4K,4Q,4J)
    To reach 4 points, we have 5 ways: (0A+0K+0Q+4J),(0A+0K+1Q+2J),(0A+0K+2Q+0J),(0A+1K+0 Q+1J),(1A+0K+0Q+0J)
    Or another example to reach 7 points we have 9 ways: (0A+0K+2Q+3J),(0A+0K+3Q+1J),(0A+1K+0Q+4J),(0A+1K+1 Q+2J),(0A+1K+2Q+0J),(0A+2K+0Q+1J),(1A+0K+0Q+3J),(1 A+0K+1Q+1J),(1A+1K+0Q+0J)

    How can I get a formula to find the number of ways to reach a specific point "n".

    Regards.
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  2. #2
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    Re: Card game: number of combinations

    Quote Originally Posted by billobillo View Post
    In a card game an Ace = 4 points, king=3 points, queen=2 points and jack=1 point.
    Now we have only 16 cards that have points to deal with(4A,4K,4Q,4J)
    To reach 4 points, we have 5 ways: (0A+0K+0Q+4J),(0A+0K+1Q+2J),(0A+0K+2Q+0J),(0A+1K+0 Q+1J),(1A+0K+0Q+0J)
    Or another example to reach 7 points we have 9 ways: (0A+0K+2Q+3J),(0A+0K+3Q+1J),(0A+1K+0Q+4J),(0A+1K+1 Q+2J),(0A+1K+2Q+0J),(0A+2K+0Q+1J),(1A+0K+0Q+3J),(1 A+0K+1Q+1J),(1A+1K+0Q+0J)
    If you expand \prod\limits_{j = 1}^4 {\left( {\sum\limits_{k = 0}^4 {{x^{jk}}} } \right)} the coefficients of the x terms answer your question. That is both 5x^4~\&~9x^7 are in the expansion.
    I use the CAS MathCad to eapand.
    Here is WolframAlpha .
    Thanks from billobillo
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  3. #3
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    Re: Card game: number of combinations

    Thank you for your time,nice solution, but is there a way to get the 5 or 9 without expanding the whole formula?
    A combination formula for example?
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    Re: Card game: number of combinations

    Quote Originally Posted by billobillo View Post
    Is there a way to get the 5 or 9 without expanding the whole formula?
    A combination formula for example?
    In your OP you requested "How can I get a formula to find the number of ways to reach a specific point "n"."
    That is what I did. So far as I know there is no formula for particular terms.
    These problems are solved by generating functions.
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  5. #5
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    Re: Card game: number of combinations

    Well thanks again for your help, I was having a hard time for finding a solution, and your solution is so helpful.
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