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This is purely for interest.

So I received from my bank a randomly generated (?) login code which is a string of 9 numerical digits (0-9). 5 of these 9 digits were the same number (but not in any particular order).

I am trying to work out the probability in any 9 digit number of 5 of these digits being the same. (and generalising this).

I have tried to work this out as I believe the answer to be of the form

$$\binom{9}{5}0.1^50.9^410$$

But my formula doesn't seem to work when I generalise this to the probability of getting say, 2 digits the same within the string of 9 - as then it give a nonsensical answer.

Am I going wrong somewhere? I am more interested in understanding the theory rather than just the actual solution.

Sven Hohenstein
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Tarquinius
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  • You are on the right track I would say, with [binomial probabilities](https://en.wikipedia.org/wiki/Binomial_distribution). – GeoMatt22 Oct 14 '16 at 20:06
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    This is a birthday problem in which we're looking for a set of 5 people with the same birthday instead of 2. (And there are 9 people in the room and only 9 possible birthdates.) – Kodiologist Oct 14 '16 at 20:24
  • @Kodiologist With digits 0 to 9, are there not 10 possible birthdays? – Silverfish Oct 15 '16 at 07:40
  • While the problem to be solved is a duplicate, I'm unsure about the question being one - mostly because this question offers a solution and asks why it's incorrect. I think it might be better to leave this Q open with an answer that explains why the OP's logic is faulty ... that answer might just link to the other thread to show how to do it properly, but explaining the logical error here doesn't seem to be a duplicate as the misconception that produced it was not dealt with (as far as I can see) on the other thread – Silverfish Oct 15 '16 at 07:45
  • @Silverfish "With digits 0 to 9, are there not 10 possible birthdays?" — Yes, my mistake. – Kodiologist Oct 15 '16 at 14:16

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