Permutation with circular arrangement

by Andrei Lenedin   Last Updated July 12, 2019 07:20 AM

In how many ways can 5 men and 5 women be arranged, if two particular women must not be next to a particular man?

May anyone explain why the answer is 864:

If the separated man $O$ is the starting point and the separated women $W_1,W_2$ are to choose any seat except the two seats beside $O$, so they would have 7 seats to choose from, implying $P^7_2$ arrangements. Then the other men and women would have $P^4_4 P^3_3$ choices respectively. Therefore the total arrangements would be $P^4_4 \times P^3_3 \times P^7_2$, however my logic is incorrect since the actual answer is $P^4_4 \times P^3_3 \times P^3_2$ but this suggests $W_1,W_2$ only have 3 seats to choose instead of 7, why is it so?

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