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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- Sep 9th 2007, 09:14 AMMr_Greenhow 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?

- Sep 9th 2007, 09:40 AMSoroban
Hello, Mr_Green!

You can talk your way through this one . . .

Quote:

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.