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?
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.