# in how many ways can the letters of the word FRACTION be arranged so that no two vowels are together.

FRACTION

vowels = AIO, let k

FRCTNK

6! => 6X5X4X3X2X1= 720

so, 720 ways can be arranged without two vowels are together.

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total ways in which the letters of the word FRACTION can be arranged = 8! = 40320
vowels : A I O ( let us consider them as 1 letter instead of 3 letters and let this new letter be * )
consonants : F R C T N
now the new word thus formed will be : * F R C T N
the letters of this new word can be arranged in 6! ways. not only that * can arrange itself in 3! ways
so the total ways in which all the vowels are together = 6! x 3! = 4320
so total words that can be formed so that no two vowels are together = 40320 - 4320 = 36000
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