Determine the number of arrangements of the letters of the word FLIBBERTIGIBBET such that there is...

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Determine the number of arrangements of the letters of the word FLIBBERTIGIBBET such that there is exactly one vowel (and any number of consonants) between the first and second B, exactly one vowel (and any number of consonants) between the second and third B, and exactly one vowel (and any number of consonants) between the third and fourth B. For example, FTIBREBIGBILBET satisfies the restriction but FTBIREBIGBILBET does not since there are two vowels between the first two B's and FTIBREBIBGBILET does not since there are no vowels between the third and fourth B. Justify your answer. Determine the number of arrangements of the letters of the word FLIBBERTIGIBBET such that there is exactly one vowel (and any number of consonants) between the first and second B, exactly one vowel (and any number of consonants) between the second and third B, and exactly one vowel (and any number of consonants) between the third and fourth B. For example, FTIBREBIGBILBET satisfies the restriction but FTBIREBIGBILBET does not since there are two vowels between the first two B's and FTIBREBIBGBILET does not since there are no vowels between the third and fourth B. Justify your answer.