# Math Help - how many keys?

1. ## 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?

2. 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.