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[tex] \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(2 \times 1)(2 \times 1)(2 \times 1)} [/tex]
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Using the arrangements formula, it is found that the number of distinguishable permutations of the letters in the word vaccination is of 2,494,800.

What is the arrangements formula?

The number of possible arrangements of n elements is the factorial of n, that is:

[tex]A_n = n![/tex]

If there are repeating elements, repeating [tex]n_1, n_2, \cdots, n_n[/tex] times, we have that:

[tex]A_n^{n_1, n_2, n_3} = \frac{n!}{n_1!n_2! \cdots n_n!}[/tex]

In this problem:

  • The word vaccination has 11 letters, hence n = 11.
  • a, c, n and i repeat twice, hence [tex]n_1 = n_2 = n_3 = n_4 = 2[/tex].

Thus:

[tex]A = \frac{11!}{2!2!2!2!} = 2,494,800[/tex]

The number of distinguishable permutations of the letters in the word vaccination is of 2,494,800.

More can be learned about the arrangements formula at https://brainly.com/question/10623514

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