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Math Help - how many keys?

  1. #1
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    how many keys?

    A certain key is cut for a lock with seven tumblers, each of which has 4 depths and no two consecutive tumblers can be the same depth. How many different keys can be made?
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  2. #2
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    Hello, Mr_Green!

    You can talk your way through this one . . .


    A certain key is cut for a lock with seven tumblers, each of which has 4 depths
    and no two consecutive tumblers can be the same depth.
    How many different keys can be made?

    Call the seven tumblers: A,\,B,\,C,\,D,\,E,\,F,\,G

    Tumbler A can be any of the 4 depths.
    . . It has 4 choices.

    Tumbler B must not have the same depth as A.
    . . It has 3 choices.

    Thumbler C must not have the same depth as B.
    . . It has 3 choices.

    Tumbler D must not have the same depth as C.
    . . It has 3 choices.

    . . . Get the idea?


    Therefore, there are: . 4 \times 3^6 \;=\;2916 different keys.

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