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Suppose I have a set of elements $$ S = \lbrace A,B,B,D,E \rbrace$$

Then the number of distinct permutations of size 5 is

$$\frac{5!}{2!} = 60$$

This is because $S$ contains a ‘copy’ of B.

But by definition, in a set of elements, each element is distinct. However, now I’m essentially saying “wait actually B and B aren’t distinct.”

So why isn’t the set $$ S = \lbrace A,B,D,E \rbrace$$ if the B elements are not distinct? Notice the permutations of this set are very different

$$4! = 24$$

Are there two different kinds of being distinct here?

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    $\begingroup$ You should think of $S$ as a multiset, not a set; one in which multiple copies do matter. Alternatively, put a yellow sticker on one of the $B$s and a green sticker on the other one so that they are now distinct. Count the number of permutations; then remove the stickets. Figure out how many ways the final permutation could have occurred to get the correct divisor. $\endgroup$ – Arturo Magidin Dec 17 '18 at 22:43
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    $\begingroup$ @ArturoMagidin Can you add this as an answer? I think this is what I am looking for. Perhaps expand a bit if you can. $\endgroup$ – Stan Shunpike Dec 17 '18 at 22:49
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You should see the counts as the number of outputs of two different experiments:

First we have a bag of 5 letter tiles (a la Scrabble say) with $A,B,B,D,E$ tiles. We draw 5 tiles from the bag and put them in a row in order we draw them. We have 5 draws, so $5!$ many permutations of the tiles. But in reality we cannot (in the end) distinguish the two B's (We could have marked them and then we'd have $5!$ different results): so the number of end results (sequences of 5 letters) is halved, because for every permutation $\ldots B_1\ldots B_2\ldots$ have a seocnd equivalent one $\ldots B_2\ldots B_1\ldots$.

The second experiment we have a bag with four tiles $A,B,D,E$ and counts the number of possible permutations from a draw of $4$ letters put in a row, so there we count sequences of $4$ letters, not of $5$. Hence the difference in counts.

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take a simpler case $\{A,B,B\}$ vs $\{A,B\}$. First one will have $ABB,BBA,BAB$, however the second one only $AB,BA$. Note that when $BB$ are next to each other it can be mapped to the single instance case one-to-one, but not when there are separated.

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