Question

Find the number of different ways of arranging letters in the word PLATOON if the two O’s are never together

Answer 1

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