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Problem: consider a three code $abc$ such that $a,b, c$ are assigned one of the values from the numbers $1,2,3,4,5.$

i) find the total number of possible codes where each value can be repeated (example: $121$ or $444$).

ii) assuming no value is repeated.

My thoughts, for i) isn't it just $5^3$ as repetition is allowed? Also for ii) since no repetition then wouldn't it be just $5\times 4 \times 3$ ? I am not entirely sure about. my answers so was checking, I will appreciate the help if the answers are wrong

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    $\begingroup$ Yes, these are both correct. $\endgroup$
    – JMoravitz
    Commented Jan 12, 2023 at 15:22

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Your first answer is correct, since there are five numbers to choose from for every slot. If no value is repeated, then you have five numbers to choose for the first slot, then one less for the second, and finally two more less for the third, so your second answer is also right.

As an interesting side note, what if the codes MUST have repetition? To do that, we take the number of possible permutations with or without repetition ($5^3=125$), then subtract the number of codes that don't have repetition ($5\times4\times3=60$), so the number of codes that have repetition is $125-60=65$.

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