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In how many ways can the letters of the word PERMUTATIONS be arranged if the, vowels are all together?

Q.11.    In how many ways can the letters of the word PERMUTATIONS be arranged if the

           (ii) vowels are all together?

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There are 5 vowels in word PERMUTATIONS  and each appears once.

Since all 5 vowels are to occur together so can be treated as 1 object.

The single object with the remaining 7 objects will be 8 objects.

The 8 objects in which 2 T's repeat can be arranged as 

                                                                                         =\frac{8!}{2!} ways.

These 5 vowels can also be arranged in 5! ways.

Hence, using the multiplication principle, the required number of arrangements are 

                                                                                                                                 =\frac{8!}{2!}\times 5!=2419200   ways.

 

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