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Thread: 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: $\displaystyle A,\,B,\,C,\,D,\,E,\,F,\,G$

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

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

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

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

    . . . Get the idea?


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

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