Question

In how many ways can 4 men and 3 ladies be arranged at a round table if 3 ladies (i) always sit together (ii) never sit together?

Answer 1

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Case 1 - All women sit together.

Take women as 1 group. So we now have 4 men and 1 group .

These 5 men can arranged in round table in (5-1)! = 4! = 24 ways.

But women can be further arranged in group in 3!= 6 ways.

Therefore total number of ways is equal to 24 × 6 = 144 ways .

Case 2- No women sit together.

As number of women is less than number of men , two of the men will be in between a pair of women. So fix the seats of women.

Women can be arranged in 3!= 6 ways.

Now select two men.

They can be selected in C(4,2) = 6 ways and arranged in 2! = 2 ways.

Consider these 2 men as a single man.

Now we have 3 men and they can be arranged in the round table in (3-1)! = 2! = 2 ways .

Therefore total number of ways = 6 × 6 × 2 × 2 = 144 ways .

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