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Question
In how many ways can the letters of the word PERMUTATIONS be arranged if the there are always 4 letters between P and S?
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Solution
There should be 4 letters between P and S.
Suppose the places of 12 letters have been named 1, 2, 3,... 12.
1 2 3 4 5 6 7 8 9 10 11 12
Thus, P can be placed at positions 1, 2, 3, 4, 5, 6, 7 and S can be placed at positions 6, 7, 8, 9, 10, 11, 12.
⇒ P and S can be placed at 7 places.
Similarly, S and P can be placed in 7 places.
P and S or S and P can be arranged in 7 + 7 = 14 ways.
The remaining 10 letters can be arranged in `(10!)/(2!)` ways.
∴ Number of words when there are 4 letters between P and S = `(10!)/(2!) xx 14 = 10! xx 7`
= 25401600.
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