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:

Tumbler can beanyof the 4 depths.

. . It has 4 choices.

Tumbler must not have the same depth as .

. . It has 3 choices.

Thumbler must not have the same depth as .

. . It has 3 choices.

Tumbler must not have the same depth as .

. . It has 3 choices.

. . . Get the idea?

Therefore, there are: . different keys.