In how many different ways can 4 ladies and 4 gentlemen be

Question

In how many different ways can 4 ladies and 4 gentlemen be seated at a round table so that all ladies sit together?

A. 70
B. 288
C. 576
D. 10,080
E. 20,160

Bunuel · Accepted Answer

Glue the ladies together so that they create one unit, so we would have 5 units: {M1}, {M2}, {M3}, {M4}, and {W1,W2,W3,W4} --> # of different arrangements of  n  objects around the table (circular arrangements) is  (n-1)! , so our 5 objects can be arranged in  (5-1)!=4!  ways.

On the other hand 4 women within their unit also can be arranged in 4! ways --> total  4!*4!=576 .

Answer: C.

vinay.kaipra · Answer

Good question... Yet another good explanation from the Master.....

shrouded1 · Answer

Treat the 4 ladies as one object, now you have 5 objects to arrange around a table (m1,m2,m3,m4,women). This can be done in (5-1)! ways
And there are 4! ways to arrange ladies among themselves

Answer = (4!)^2 = 576 or C

usre123 · Answer

For concept's sake, if we were to do this the opposite way, how would we do it? Say we have (8-1)! of arranging without any conditions. Then it should be 7! - number of ways 2 women can sit together - number of ways three can sit together.

so for number of ways two can sit together I get: (4-1)! and then 4C3 (in how many ways can we place 3 women in 4 slots, since I tied two together * 2)

Number of ways 3 can sit together= seat the men in (4-1)! ways. * 4C2 (in how many ways can two women be placed in 4 slots, since I tied three women together this time)*3! (for the number of arrangements of three women ties together)

This doesn't give me the correct answer. Where have I gone wrong?

CEdward · Answer

_ _ _ _ _ _ _ _ <---- 8 spots

Combination:
Since the four women must be together there is 4C4 ways we can choose seats for them.

Permutation:
Among the women, there are 4! ways we can arrange them.
Likewise among men there are 4! ways that they can be arranged

4! x 4! = 576.

Answer is C.

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In how many different ways can 4 ladies and 4 gentlemen be

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In how many different ways can 4 ladies and 4 gentlemen be

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