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Question
In how many ways four men and three women may sit around a round table if all the womeni) always sit tohetherii) never sit together
i) always sit tohether
ii) never sit together
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Solution
Case 1: If all three women always sit together
We can club the 3 women in one group and the number of arrangements of this group will be 3!
Now, 4 men and 1 group of women can be arranged arround a round table =(5−1)!
Hence, Total arrangements =4!3!=144.
Now, case 2: If all three women never sit together
Total arrangements without any constraints =6!
From case 1, we know the arrangements if women always sit together.
Hence, Total arrangements =6!−144=720−144=576=6!−4!3!=576
Case 1: If all three women always sit together
We can club the 3 women in one group and the number of arrangements of this group will be 3!
Now, 4 men and 1 group of women can be arranged arround a round table =(5−1)!
Hence, Total arrangements =4!3!=144.
Now, case 2: If all three women never sit together
Total arrangements without any constraints =6!
From case 1, we know the arrangements if women always sit together.
Hence, Total arrangements =6!−144=720−144=576=6!−4!3!=576
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