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Question
In how many ways can the letters of the word PERMUTATIONS be arranged if the vowels are all together.
Solution
In the word PERMUTATIONS, there are 2 Ts and all the other letters appear only once.
There are 5 vowels in the given word, each appearing only once.
Since they have to always occur together, they are treated as a single object for the time being. This single object together with the remaining 7 objects will account for 8 objects. These 8 objects in which there are 2 Ts can be arranged in `(8!)/(2!)` ways
Corresponding to each of these arrangements, the 5 different vowels can be arranged in 5! ways.
Therefore, by multiplication principle, required number of arrangements in this case
= `(8!)/(2!) xx 5! = (40320 xx 120)/2`
= 2419200.
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