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From Mathematical Puzzles by Geoffrey Mott-Smith: If a box of anagram letters contains nine M's, twenty-eight I's, twenty-four S's and eight P's, in how many ways can you pick out and arrange letters to make MISSISSIPPI?

The solution in the book is 508,722,691,276,800.

This is what I get:

$\frac{9 \times 28 \times 24 \times 23 \times 27 \times 22 \times 21 \times 26 \times 8 \times 7 \times 25}{2 \times 4! \times 4!}= 54,826,972,200$

That is, I multiply the number of available letters for each letter in the word, ensuring to decrease the count by one when a given letter has already been used, and then dividing by the number of occurrences of each repeating letter. What am I doing wrong?

EDIT I apologize for the sloppiness, the problem statement is meant to read "nine M's"

Here are photos taken from the book:

problem statement

solution

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    $\begingroup$ Are you sure you want just a single P? $\endgroup$ – Lukas Kofler Aug 13 at 14:32
  • $\begingroup$ sorry, typo, will fix $\endgroup$ – user695532 Aug 13 at 14:35
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    $\begingroup$ "...letters contains eight nine* M's" What is meant here? Do you mean eighty-nine M's? I.e. $89$ M's? $\endgroup$ – JMoravitz Aug 13 at 14:37
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I will assume that there are nine $M$'s, not eighty-nine $M$'s or some other number.

I will further assume that each tile is distinct, with its own unique id, and that two arrangements will be considered different iff there is at least one position in the first arrangement whose tile has a different id than the same position in the other arrangement.

For example $M_1 I_{10} S_{50}S_{51}\dots$ will be a different arrangement than $M_1 I_{10}S_{51}S_{50}\dots$ since the tile in the third position in the first arrangement is different than the tile in the third position in the second arrangement. This is despite them both having the same letter appearing, what is different is the id of the tile.

We see then that your attempt is very close to being correct... but you divided when you had no reason to. The answer would have simply been

$$9×28×24×23×27×22×21×26×8×7×25 = 63,160,671,974,400$$

The division, if we would have performed it, would have been in order to "forget" which tile was used in which position, your answer with the division effectively being how many ways we can pull tiles simultaneously such that we pulled one $M$, four $I$'s, etc... but this calculation would not take into consideration the number of ways of arranging the tiles, instead only taking into consideration the number of ways of selecting the tiles.

If we did not care about which id for a tile was used in each position... then there is very simply only one possible arrangement that we could have such that the letters form the word mississippi... that would be the arrangement mississippi...

Now... as for the book's answer, we can look at the factorization of it for clues as to what it might be an answer for. We see that $508722691276800$ factors as:

$$2^{15}\times 3^6\times 5^2\times 7\times 11\times 13\times 23\times 37$$

I see no possible interpretation of the problem with the numbers as given that could allow for $37$ to be a factor of the answer and so suspect that you copied the answer incorrectly, you copied the problem incorrectly, or the answer itself was incorrect.

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