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# Seven family members are seated around their circular dinner

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Re: Seven family members are seated around their circular dinner [#permalink]
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Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

Lets consider Michael and Bobby as one individual and fix their position so that all the members do NOT move together while they remain in same order relatively

Now after fixing Michael and Bobby we have 5 other member left to change their positions among themselves which can change positions in 5! ways

but Michael and Bobby and exchange positions between the two in 2! ways'

Hence, Total ways of different arrangements = 5!*2! = 120*2 = 240

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Re: Seven family members are seated around their circular dinner [#permalink]
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Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

If Bobby and Michael must sit next to each other, we treat them as a single entity, and that leaves us with 6 total spots to arrange. Using the circular permutations formula (n - 1)!, we have

(6 - 1)! = 5! = 120 ways to arrange the family members with Bobby and Michael together.

However, we also must include the number of ways to arrange Bobby and Michael, which is 2P2 = 2! = 2.

So, in total, we have:

(6 - 1)! * 2! = 120 x 2 = 240

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Re: Seven family members are seated around their circular dinner [#permalink]
Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

$$(6 - 1 )!*2!$$

$$= 5!*2!$$

$$= 240$$, Answer must be (B)
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Re: Seven family members are seated around their circular dinner [#permalink]
Bunuel wrote:
Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

Glue Michael and Bobby so that they create one unit, so we would have total of 6 units: {1}{2}{3}{4}{5}{MB} --> # of different arrangements of $$n$$ objects around the table (circular arrangements) is is $$(n-1)!$$, so our 6 objects can be arranged in $$(6-1)!=5!$$.

On the other hand Michael and Bobby in 2! ways --> total $$5!*2!=240$$.

Similar question also posted by you: https://gmatclub.com/forum/ways-to-sit-a ... 02187.html

I also did in the same manner and this method is very quick.
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Re: Seven family members are seated around their circular dinner [#permalink]
Using the circular permutation concept,

Without any restriction, no. of ways arrangement = 6!

With restriction, no. of ways arrangement = (6-1)!*2!=5!*2!=120*2=240

Hence B
Re: Seven family members are seated around their circular dinner [#permalink]
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