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 Question from Aeriel, a student: A lock works by having a key turn a sequence of tumblers. Consecutive tumblers have different heights, and in order to unlock the lock, the sequence of heights on the key must exactly match those of the lock. The picture below shows a lock with four tumblers. How many different keys can be made for a lock that has 7 tumblers with 5 possible heights each?

Hi Aeriel,

Suppose the lock had only 1 tumbler then, since it has 5 possible heights, 5 different keys are possible. Now suppose you add a tumbler then regardless of the height you have set for the first tumbler you have 5 choices for the height of the second tumbler. Hence for a 2 tumbler lock there are $5 \times 5 = 5^2$ different possible keys.

Now suppose you add a third tumbler. Again regardless of the height you have set for the first two tumblers you have 5 choices for the height of the third tumbler. Hence for a 3 tumbler lock there are $5^2 \times 5 = 5^3$ different possible keys.

Continue,
Penny

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