The number of different permutations of all the letters of the word 'PERMUTATION' such that any two consecutive letters in the arrangement are neither both vowels nor both identical is

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The letters other than vowels are : PRMTTN

Number of permutations with no two vowels together is $\frac{6 !}{2 !} \times^{7} C_{5} \times 5!$

Further among these permutations the number of cases in which T's are together is $5! \times^{6} C_{5} \times 5!$

So the required number $= \frac{6 !}{2 !}^{7} C_{5} \times 5! - 5! \times^{6} C_{5} \times 5! = 57 \times (5!)^{2}$

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