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![[#45] 2 Min. Challenge: Six People](/forum/styles/gmatclub_light/imageset/icon_post_target_unread.svg "[#45] 2 Min. Challenge: Six People"){kind=icon}
11 May 2004, 18:57
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Level - Intermediate
Time yourself and solve under 2 minutes. Please explain your solution.
Six people are going to sit in a row on a bench. Romeo wants to sit next to Juliet. Caesar does not want to sit next to Brutus. Homer and Pierre can sit anywhere. How many ways can these six people be seated?
a. 24
b. 56
c. 120
d. 144
e.200
Time yourself and solve under 2 minutes. Please explain your solution.
Six people are going to sit in a row on a bench. Romeo wants to sit next to Juliet. Caesar does not want to sit next to Brutus. Homer and Pierre can sit anywhere. How many ways can these six people be seated?
a. 24
b. 56
c. 120
d. 144
e.200
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12 May 2004, 00:38
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Praetorian
If "next" means that A and B do not want to sit closely (in (a,b), or in (b,a) order), then:
we have 5 ways to sit for Romeo & Juliet, and there are 2 pairs of symmetric allocations ((r,g,*,*,*,*) and (*,*,*,*,r,g), for example)
2*{2*[2 + 1 + 1 + 2]*2 + 2*[3 + 2 + 1 + 2]*2} //2 before { stand for 2 symmetric pairs// + 2*[2 + 2 + 2 + 2]*2 //central allocation - (*,*,r,g,*,*)// = 144.
After we fixed (r,g), we consider Caesar - how many allocations with Brut he can make? Finally, we *2, and again *2 because the order of R and G, and the order of H and P doesn't matter.
D, 144.
12 May 2004, 06:56
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Combinations with Juliet and Romeo together = 2 * 5! = 240
Out of these the combinations with caesar and and Brutus together are invalid = 2 * 2 * 4! = 96
Total arrangements = 240-96 = 144
Out of these the combinations with caesar and and Brutus together are invalid = 2 * 2 * 4! = 96
Total arrangements = 240-96 = 144
