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In how many different ways can 4 ladies and 4 gentlemen be [#permalink]
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In how many different ways can 4 ladies and 4 gentlemen be seated at a round table so that all ladies sit together?A. 70B. 288C. 576D. 10,080
E. 20,160
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Re: Ways to sit around the table [#permalink]
eladshush wrote:
In how many different ways can 4 ladies and 4 gentlemen be seated at a round table so that all ladies sit together?A. 70B. 288C. 576D. 10,080
E. 20,160
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. _________________
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Re: Ways to sit around the table [#permalink]
Good question... Yet another good explanation from the Master.....
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Re: Ways to sit around the table [#permalink]
eladshush wrote:
In how many different ways can 4 ladies and 4 gentlemen be seated at a round table so that all ladies sit together?A. 70B. 288C. 576D. 10,080
E. 20,160
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)! waysAnd there are 4! ways to arrange ladies among themselvesAnswer = (4!)^2 = 576 or C
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Re: In how many different ways can 4 ladies and 4 gentlemen be [#permalink]
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?
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Re: In how many different ways can 4 ladies and 4 gentlemen be [#permalink]
_ _ _ _ _ _ _ _ <---- 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 arranged4! x 4! = 576.Answer is C. _________________
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Re: In how many different ways can 4 ladies and 4 gentlemen be [#permalink]
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Re: In how many different ways can 4 ladies and 4 gentlemen be [#permalink]
24 Sep 2022, 15:00