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 Question from Sophia: How many 4 digit codes can you get out of a numbered key pad ?

Hi Sophia,

I assume order is important, that is 1234 and 4321 are different codes, but can you repeat digits? For example is 2454 a permissible code?

If the answer is yes then think about entering a code. You have 10 choices for the first digit you enter, 0 to 9. Regardless of the digit you chose you can extend your code to two digits by again choosing any of the 10 digits and hence you have $10 \times 10$ choices for a two digit code. Again regardless of the Iwo digits you chose you can extend your code to three digits by again choosing any of the 10 digits and hence you have $10 \times 10 \times 10$ choices for a three digit code. In a similar fashion there are $10 \times 10 \times 10 \times 10 = 10,000$ possible 4 digit codes.

Before we go on to the situation where repeats are not allowed look at this first case again. Think of the codes as numbers starting at zero, written 0000, and then one written 0001, up to nine written 0009 and then ten, written 0010 and so on until 9999. Thus you have as possible combinations the 10,000 numbers from 1 to 9999.

What if repeats are not allowed? Again you have 10 choices for the first digit you enter, 0 to 9. Regardless of the digit you chose you can extend your code to two digits by again choosing any of the 9 remaining digits and hence you have $10 \times 9$ choices for a two digit code. Hopefully now you see that there are $10 \times 9 \times 8 \times 7$ possible four digit codes.

Penny

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