A certain key is cut for a lock with six tumblers, each of which has five depths and no two consecutive tumblers can be the same depth. How many different keys can be made?
2916
To find the number of different keys that can be made, we need to consider the possible options for each tumbler, while taking into account the restriction that no two consecutive tumblers can be the same depth.
Let's break down the problem into steps:
Step 1: Choose the depth for the first tumbler.
Since there are five possible depths for each tumbler, we have five options for the first tumbler.
Step 2: Choose the depth for the second tumbler.
Since no two consecutive tumblers can be the same depth, we have four options for the second tumbler.
Step 3: Choose the depth for the third tumbler.
Again, we have four options for the third tumbler, as it cannot have the same depth as the second tumbler.
Step 4: Repeat Steps 2 and 3 for the next three tumblers.
For the fourth, fifth, and sixth tumblers, we still have four options for each, as they cannot have the same depth as the previous tumbler.
Step 5: Multiply the number of options for each step together.
To calculate the total number of different keys, we need to multiply the number of options for each step together:
5 options for the first tumbler × 4 options for the second tumbler × 4 options for the third tumbler × 4 options for the fourth tumbler × 4 options for the fifth tumbler × 4 options for the sixth tumbler.
Using simple multiplication, we can find the answer:
5 × 4 × 4 × 4 × 4 × 4 = 20,480.
Therefore, there are 20,480 different keys that can be made for the lock with six tumblers.