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 Question from Judy, a student: Hello: Can you please explain why the answer to the following question is 10^6? What is the number of possible six-digit passwords when using the digits 0 through 9, with repetition allowed. Thank you.

Hi Judy,

Think about the simpler question

What is the number of possible one-digit passwords when using the digits 0 through 9, with repetition allowed?

The answer is clear, there are $10$ such passwords and you can write them all down, $0, 1, 2, \cdot \cdot \cdot, 9.$

Now suppose the question is

What is the number of possible two-digit passwords when using the digits 0 through 9, with repetition allowed?

Each of the one-digit passwords you already have can be extended to a two-digit password by adding a second digit, so for example $0$ extends to $00, 01, 02, \cdot \cdot \cdot, 09$ and $1$ extends to $10, 11, 12, \cdot \cdot \cdot, 19.$ Thus each one-digit password can be extended to $10$ two-digit passwords and hence there are $10 \times 10 = 10^2$ two-digit passwords.

What is the number of possible three-digit passwords when using the digits 0 through 9, with repetition allowed?

This time each of the two-digit password you already have can be extended to a three-digit password by adding a third digit. Again, in each case there are $10$ choices for the third digit so the number of three-digit passwords is $10$ times what you already had, that is $10^2 \times 10 = 10^3 .$

Repeat this process for four, five and six-digit passwords.

I hope this helps,
Penny

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