Final answer:
The number of ways to arrange the letters in 'antidisestablishmentarianism,' accounting for repeating letters, is calculated using permutations of a multiset. The exact number is obtained by dividing 28! by the factorials of the counts of each repeating letter (4! for 'a', 5! for 'i', etc.).
Step-by-step explanation:
To determine how many ways there are to arrange the letters in the word "antidisestablishmentarianism," taking into account the identical letters, we have to use the formula for permutations of a multiset. The total number of ways to arrange a set of items is factorial of the number of items, represented by n!. However, if some items are identical, we divide the factorial of the total number by the product of factorials of the number of each set of identical items.
The word "antidisestablishmentarianism" has the following letter counts: a-4, i-5, n-3, t-2, s-4, e-2, b-1, l-1, h-1, m-1, r-1. Let's calculate the permutations:
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- Total letters: 28
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- Identical letters: a, i, n, t, s, and e
The formula we'll use is:
\( \frac{28!}{4! \times 5! \times 3! \times 2! \times 4! \times 2!} \)
Calculating this gives us the total number of unique arrangements of the letters in "antidisestablishmentarianism." You can use a factorial calculator or computing software to get the exact figure, which is a very large number indeed, much greater than 500.