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Math Help - Fundemental counting question help?

  1. #1
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    Fundemental counting question help?

    Morse cose translates letters, digits, and punctuation mark into sequences of dots and dashes, with a maximum mixture of five dots and/or dashes per character. What is the maximum number of Morse code sequences, each representing a character, that can be created.

    The answer should be 62.

    Maybe it's because I don't exactly understand Morse code, so I don't exactly know how to do the question? Can someone show me the steps? Thanks!
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  2. #2
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    Re: Fundemental counting question help?

    Quote Originally Posted by wabbt View Post
    Morse cose translates letters, digits, and punctuation mark into sequences of dots and dashes, with a maximum mixture of five dots and/or dashes per character. What is the maximum number of Morse code sequences, each representing a character, that can be created. The answer should be 62.
    Note that \sum\limits_{k = 1}^5 {2^k }  = 62
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  3. #3
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    Re: Fundemental counting question help?

    If the maximum length of a Morse code sequence is 5 characters (a dot or dash), then there are 5 different Morse code sequences lengths (length 1, length 2, length 3, length 4, length 5). The total number of Morse code sequences is the sum of the number of Morse code sequences for each length.

    Consider a Morse code sequence of length one (i.e. there is exactly one character). There are 2 possible sequences: a dot or a dash.

    Now consider a Morse code sequence of length two (i.e. there are exactly two characters). There are 2 choices (a dot or dash) for the first character and 2 choices (a dot or dash) for the second character. Therefore, there are 2 \times 2 = 2^2 = 4 possible Morse code sequences of length two.

    Consider the remaining cases if you don't see the pattern yet.

    Once you know the number of Morse code sequences for each length, add them up to get the total number of Morse code sequences: 2 + 2^2 + 2^3 + 2^4 + 2^5 = \sum_{k=1}^5 2^k (something Plato already mentioned).

    If you're more familiar with binary code, note this problem is identical to how many messages of length five can made using only 0s or 1s.
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