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Question
Find the number of different ways of arranging letters in the word PLATOON if the two O’s are never together.
Solution
When the two O’s are never together:
Let us arrange the other 5 letters first, which can be done in 5! = 120 ways.
The letters P, L, A, T, N create 6 gaps, in which O’s are arranged.
∴ Two O’s in 6 gaps can be arranged in `(""^6"P"_2)/(2!)` ways
= `((6!)/((6-2)!))/(2!)` ways
= `(6xx5xx4!)/(4!xx2xx1)` ways
= 3 × 5 ways
= 15 ways
∴ Total number of arrangements if the two O’s are never together = 120 × 15 = 1800
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