In how many ways can the letters of the word PERMUTATIONS be arranged if the.i) Words start with P and end with S.ii) vowels are all togetheriii) there are always 4 letters between P and S

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i) There are total 12 letters in the word PERMUTATIONS, with T repeated twice.If P and S are fixed at the extreme ends (P at the left end and S at the right end), then 10 letters are left.These ten letters with T occurring twice can be arranged in \cfrac{10!}{2!}.Hence,  required number of arrangements  =\cfrac {10!}{2!}=1814400ii) There are total 12 letters in the word PERMUTATIONS, with T repeated twice.Number of vowels in the given word are 5.Since vowels have to always occur together, so they are considered as a single object . This single object (letter) together with the remaining 7 letters will give us 8 objects (letters).  These 8 objects in which there are 2 T's can be arranged in \cfrac{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 =\cfrac {8!}{2!} 	imes 5!= 2419200iii) There are 12 letters in word PERMUTATIONS. T occurs twice .Here, P and S are fixed. These P and S can interchange their position. Hence, P and S can be arranged in 2! ways.Since, P and S are fixed, there are 10 letters left , where T occurs twice .From these 10 letters, 4 letters can be chosen in \cfrac{^{10}C_4}{2!} ways.Now since,  4 letters are between P and S, so these six letters can be considered as a single object (letter). These four letter (between P and S )can be arranged in 4! ways.Now, the remaining 6 letters and 1 object i.e. total 7 can be arranged in 7! ways.Hence, required number of ways =2!\cfrac {^{10}C_4}{2!}\,  4! \,  7!                                                        =25401600

Question Text	In how many ways can the letters of the word $PERM U T A T I ONS$ be arranged if the.i) Words start with $P$ and end with $S$ .ii) vowels are all togetheriii) there are always $4$ letters between $P$ and $S$
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In how many ways can the letters of the word $PERM U T A T I ONS$ be arranged if the.i) Words start with $P$ and end with $S$ .ii) vowels are all togetheriii) there are always $4$ letters between $P$ and $S$

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