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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 can take their places in 6P2 ways.
But ‘O’ repeats 2 times.
∴ Two O’s can be arranged in `(""^6"P"_2)/(2!)`
= `((6!)/((6 - 2)!))/(2!)`
= `(6 xx 5 xx 4!)/(4! xx 2 xx 1)`
= 3 × 5
= 15 ways
∴ Total number of arrangements if the two O’s are never together = 120 × 15 = 1800
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