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Thread: Total Possible Outcomes Help

  1. #1
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    Question Total Possible Outcomes Help

    Hoping someone smarter than me can help me come up with the right answer here.

    Setup:
    I have a device with eight (8) lights
    Each light can light up one of seven (7) colors
    When on, each light can flash fast, slow, or solid (3)
    Off is also a possibility, but if any of the sets are off, all are off

    How many possible ways can this device be configured?

    Examples:
    Lamp 1 Lamp 2 Lamp 3 Lamp 4 Lamp 5 Lamp 6 Lamp 7 Lamp 8
    Red Off Off Off Off Off Off Off
    Solid Off Off Off Off Off Off Off

    Lamp 1 Lamp 2 Lamp 3 Lamp 4 Lamp 5 Lamp 6 Lamp 7 Lamp 8
    Off Red Off Off Off Off Off Off
    Off Solid Off Off Off Off Off Off

    Lamp 1 Lamp 2 Lamp 3 Lamp 4 Lamp 5 Lamp 6 Lamp 7 Lamp 8
    Red Red Off Off Off Off Off Off
    Fast Fast Off Off Off Off Off Off
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  2. #2
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    Re: Total Possible Outcomes Help

    The "fundamental law of counting": if A can occur in n ways and, for each of those ways B can occur in m ways, then together they can occur in mn ways.

    Any of 8 lamps, each in any of 7 colors, which can flash in any of 3 ways, has 8*7*3 possible combinations.
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  3. #3
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    Re: Total Possible Outcomes Help

    Quote Originally Posted by HallsofIvy View Post
    The "fundamental law of counting": if A can occur in n ways and, for each of those ways B can occur in m ways, then together they can occur in mn ways.

    Any of 8 lamps, each in any of 7 colors, which can flash in any of 3 ways, has 8*7*3 possible combinations.
    Your answer excludes the possibility of more than one light being on at a time. Anywhere from 1-8 of the lights could be active with any of these possibilities at any time.
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  4. #4
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    Re: Total Possible Outcomes Help

    There are $7\cdot 3$ possible configurations for "on" plus the "off" configuration. So, each lamp has 22 different possible configurations. Since the configuration for one lamp does not depend on the configuration for another, there are $22^8=54,875,873,536$ possible configurations total.
    Last edited by SlipEternal; Aug 11th 2017 at 11:53 AM.
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  5. #5
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    Re: Total Possible Outcomes Help

    Quote Originally Posted by SlipEternal View Post
    There are $7\cdot 3$ possible configurations for "on" plus the "off" configuration. So, each lamp has 22 different possible configurations. Since the configuration for one lamp does not depend on the configuration for another, there are $22^8=54,875,873,536$ possible configurations total.
    That's a considerably higher number than I was expecting. It makes sense if ^8 is how to express the various lamp combinations. Thank you for the insight!
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  6. #6
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    Re: Total Possible Outcomes Help

    The idea is, the first lamp has 22 possible settings. Then the second lamp has an independent set of 22 possible settings. That is $22\cdot 22$. Then the third lamp has another 22, so now we are at $22^3$. There are 8 lamps total, so if we keep multiplying by 22 for each lamp, we get $22^8$.
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