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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