In how many ways can the letters of the word VALEDICTORY be arranged so that the vowels are not separated?
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9,83,490
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9,67,680
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7,26,400
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5,63,680
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98,768
The word VALEDICTORY has 11 letters: V, L, D, C, T, R, Y (consonants) and A, E, I, O (vowels). Treating vowels as one block, we have 8 items to arrange (7 consonants + 1 block). There are 8! ways to arrange these, and 4! ways to arrange vowels within the block. Total = 8! * 4! = 40320 * 24 = 967680.
The word VALEDICTORY consists of 11 letters, comprising 4 distinct vowels (A, E, I, O) and 7 distinct consonants. Since the vowels must not be separated, group them into a single block. This vowel block can be placed among the 7 consonants in 8! different positional arrangements. The 4 vowels can be arranged within their own block in 4! ways. Multiplying these arrangement possibilities gives 8! times 4!, which is 40320 times 24, yielding 967680.