Number of permutations of the word QUADRATIC such that no two vowels and no two consonants are together, is

The word QUADRATIC has 5 consonants (Q, D, R, T, C) and 4 distinct vowels (U, A, I, A). For no two consonants to be together, the 5 consonants must be arranged first, which can be done in 5! ways. This creates 6 gaps (including the ends), and to ensure no two vowels are together, the 4 vowels must occupy 4 of these 6 distinct gaps, which inherently separates all the vowels. Because the gap selection forces strict alternation starting with consonants, the vowels are arranged in 4! ways, making the total number of valid permutations 5!4!.

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