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Inclusion-Exclusion

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How many permutations of the 26 letters of the English alphabet do not contain any of the strings fish, rat, or bird? Please show all the steps along with necessary explanations.

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There are 26! permutations of the letters of the English alphabet.

Let:
X = number of permutations containing fish
Y = number of permutations containing rat
Z = number of permutations containing bird

If the word "fish" is one symbol and there are 22 other letters, then X = 23!.

If the word "rat" is one ...

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