The probability that no consonant falls between any two vowels in the word is
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2026-04-02 04:53:48.1775105628
A word of at least 5 letters is made at random from 3 vowels and 3 consonants, all the letters being different.
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Consider two cases:
Case 1. $6$-letter words. $3!$ ways to permute $3$ vowels. $4!$ ways to permute block of $3$ vowels and $3$ consonants. $6!$ ways to permute $6$ letters in total. Hence: $$\frac{3!\cdot 4!}{6!}$$
Case 2. $5$-letter words. a) $2$ vowels and $3$ consonants; b) $3$ vowels and $3$ consonants: $$a) \ \frac{2!\cdot 4!}{5!}; \ b) \ \frac{3!\cdot 3!}{5!}.$$ Can you interpret Case 2 and finish?