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Some students can't even grok this problem statement, as they are (informationally) overloaded by the number of variables : $p, t, k, n_1, \cdots, n_k \;$. Kindly improve my picture, or draw a better picture?

I match variables to the first letter of the mathematical object. E.g. $N$umber of each $\color{fuchsia}{k}$ind $= n_\color{fuchsia}{k}$.

user1147844
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    I think the phrase "Pick $p$ permutations" is awkward and unclear. I would prefer something like "How many ways to make an ordered selection of $p$ objects?", or maybe "marbles" instead of "objects" to match the picture. Otherwise, the picture seems clear. Also, to make things simpler, you do not need to even mention $t$ (one fewer variable that way). – Mike Earnest Feb 18 '23 at 21:33
  • @MikeEarnest thanks! Doesn't "permute" mean the same thing as "make an ordered selection"? And did I need to bring up $t$ things at all? Correspondingly, did the linked original question need $n$ objects? I am dazzled if both questions carelessly introduced a redundant variable! – user1147844 Feb 19 '23 at 10:30
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    Ok, but "Pick $p$ permutations" is still nonsense. Better phrasing would be "How many ways to permute $p$ marbles?". Or, "How many $p$-permutations of marbles?" – Mike Earnest Feb 19 '23 at 17:34
  • The reason you do not need $t$ is because $t=n_1+\dots+n_k$, so it is already determined by the other variables. – Mike Earnest Feb 19 '23 at 17:53

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Imagine you are a typesetter manually placing sorts – metal blocks containing raised mirrored characters on one face – into a line. You have $n_i$ sorts of character $i$, and $k$ different characters available, so $t=\sum_{i=1}^kn_i$. Then you want to count the number of $p$-character strings you can make with your limited resources.

Parcly Taxel
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