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I'm hoping this is really simple, but I can't just seem to get it!

I have an 8 character code that consists of 3 letters [A] and 5 numbers [N]

I need to work out the possible positional/ordering combinations of those 8 characters, e.g. ANNANNNA, AANANNNN, NNNAAANN etc...

Please note I'm not concerned with the total number of possible codes (which hopefully is $26*26*26*10*10*10*10*10$ ), just the ordering combinations.

Many thanks in advance, and apologies for any bad usage of terms!

Tim Morrison
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1 Answers1

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There are two ways to look at it.

One way to look at it is that you have 8 slots to fill, and you need to choose 3 of those slots to fill with letters. That's the same problem as choosing a team of 3 people from a pool of 8 people, and there's a combinatorial function that answers that question: $\binom83 = \frac{8!}{3!(8-3)!}$

The other way to look at it is that any time you have 8 characters (say $A_1A_2A_3N_1N_2N_3N_4N_5$) to arrange in a sequence, there are $8!$ ways to do so. But since some of the characters in this case are indistinguishable, you have to divide out certain symmetries. The symmetries you have to divide out are all rearrangements of the 3 As ($3!$) and all rearrangements of the 5 Ns ($5!$). So the answer will be $\dfrac{8!}{3!5!}$

BallBoy
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