How can i calculate the number of possible PIN codes that use only numbers 0-9 and have length are there? No rules,all numbers can be same or different


closed as off-topic by TheGeekGreek, Namaste, Daniel W. Farlow, user296113, Shailesh Aug 3 '17 at 0:07

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  • $\begingroup$ What have have you tried so far? $\endgroup$ – platty Aug 2 '17 at 20:42
  • $\begingroup$ Hint: how many digits are there, and how do we count the ways a task can be done? $\endgroup$ – Sean Roberson Aug 2 '17 at 20:43
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    $\begingroup$ Both of those are wrong. I have no idea where either of your answers came from. $\endgroup$ – lulu Aug 2 '17 at 20:45
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    $\begingroup$ No, it isn't. first do it for length one though. $\endgroup$ – lulu Aug 2 '17 at 20:47
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    $\begingroup$ give an example of a PIN code of length one $\endgroup$ – Yanko Aug 2 '17 at 20:48

The key is the multiplication principle: If a task can be done in $m$ ways and another in $n$ ways, the number of ways we can do both is $mn$.

The idea is that you're choosing a digit from $0$ to $9$ and constructing a six digit PIN. We can start with the case of one digit, then two, then I'll leave it to you to extend to six.

There are only ten one-digit PINs: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. Ten digits.

What about two? We could list them out, but let's be smart: this is the same as the concatenation of two one-digit PINs. We have ten choices for the first digit and ten for the second. How many total two-digit PINs are there? By the multiplication prinicple, this is $10 \cdot 10 = 100$. There are a hundred possible 2-digit PINs.

Can you take it from here?

  • $\begingroup$ 10 to the power of 6? $\endgroup$ – Murad Aug 2 '17 at 20:50
  • $\begingroup$ Well done. That's exactly it. $\endgroup$ – Sean Roberson Aug 2 '17 at 20:50
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    $\begingroup$ wow i'm a genius $\endgroup$ – Murad Aug 2 '17 at 20:51
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    $\begingroup$ @Murad you are the most brilliant person I have ever met. $\endgroup$ – Yanko Aug 2 '17 at 20:51
  • $\begingroup$ @yanko My math teacher tells me the same thing $\endgroup$ – Murad Aug 2 '17 at 20:52

Here is another way to see this

every Pin code of lenth 6 defines a number from 0 to 999,999, that is in total one million numbers.


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