How Many Ways Can 6 People Sit on a Bench if 3 of Them Have to Sit Side by Side?
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In how many means can yous sit down 8 people on a bench if 3 of [#permalink] 
12 Mar 2012, 08:23
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In how many ways can you sit down eight people on a bench if iii of them must sit together?
A. 720
B. ii,160
C. 2,400
D. 4,320
E. xl,320
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Re: Combinations [#permalink] 12 Mar 2012, 08:35
Gavan wrote:
In how many means can you sit 8 people on a bench if 3 of them must sit together?
a) 720
b) 2,160
c) 2,400
d) 4,320
e) xl,320
i'1000 getting 'e' just that is not OA
Say 8 people are {A, B, C, D, E, F, Grand, H} and A, B and C must sit together. Consider them as 1 unit {ABC}, and so we'll take total of six units: {ABC}, {D}, {Due east}, {F}, {G}, {H}, which can be arranged in vi! ways. Now, A, B and C within their unit of measurement can exist arranged in three! ways, which gives total of 6!*iii!=4,320 different arrangements.
Answer: D.
Hope it's clear.
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Re: In how many ways tin y'all sit eight people on a bench if 3 of [#permalink] 12 Mar 2012, 08:59
thanks!
did you consider post-obit arrangements?
i: -{D},{ABC}, {E}, {F}, {G}, {H},
two: -{D},{East}, {F},{ABC}, {K}, {H},
3: -
..
Please analyze.
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Re: In how many ways can yous sit viii people on a bench if 3 of [#permalink] 12 Mar 2012, 09:xxx
Gavan wrote:
cheers!
did you consider following arrangements?
1: -{D},{ABC}, {E}, {F}, {Yard}, {H},
two: -{D},{Due east}, {F},{ABC}, {One thousand}, {H},
3: -
..
Delight clarify.
half-dozen singled-out object can exist arranged in 6! different ways, and so 6 units {ABC}, {D}, {Eastward}, {F}, {K}, {H} can exist bundled in half-dozen! dissimilar means, which takes cares of all possible cases.
Check Combinations affiliate of Math Book for more:
math-combinatorics-87345.html
Hope it helps.
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Re: In how many means can you sit 8 people on a bench if 3 of [#permalink] 29 May 2016, 22:57
Gavan wrote:
In how many ways can you sit down 8 people on a bench if 3 of them must sit down together?
A. 720
B. 2,160
C. 2,400
D. iv,320
E. 40,320
In such questions, e'er tie the person that have to sit together. So we have effectively v+"1" = 6 "persons" to arrange.
They can be arranged in half-dozen! ways.
Now the three persons can themselves be arranged in 3! means.
Total means: 6!*iii! = 4320.
D is the answer.
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Re: In how many ways can you sit down 8 people on a bench if 3 of [#permalink] 24 October 2017, 07:35
Bunuel wrote:
Gavan wrote:
In how many ways can you sit down 8 people on a bench if three of them must sit down together?
a) 720
b) 2,160
c) ii,400
d) 4,320
e) 40,320
i'm getting 'due east' just that is not OA
Say 8 people are {A, B, C, D, Due east, F, K, H} and A, B and C must sit together. Consider them as one unit {ABC}, so we'll have total of six units: {ABC}, {D}, {East}, {F}, {Thousand}, {H}, which tin be bundled in 6! ways. Now, A, B and C inside their unit tin can be arranged in iii! ways, which gives full of 6!*iii!=four,320 different arrangements.
Answer: D.
Hope it'southward clear.
quick question regarding this: Since the question doesn't specify which iii people need to sit down together, how come nosotros don't have notice ways of choosing 3 ppl out of the viii to act as unit? Like {DEF}, {AGF} etc etc.. I was thinking nosotros would have to do 8C3 x 3! x vi!... can yous tell me where I am going wrong with my logic? Thanks for your assistance!
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In how many ways can y'all sit 8 people on a bench if 3 of [#permalink] 29 Oct 2017, 11:20
Ace800 wrote:
Bunuel wrote:
Gavan wrote:
In how many ways can you sit 8 people on a bench if 3 of them must sit together?
a) 720
b) ii,160
c) 2,400
d) 4,320
e) 40,320
i'k getting 'e' but that is not OA
Say viii people are {A, B, C, D, E, F, G, H} and A, B and C must sit together. Consider them as 1 unit {ABC}, so we'll have full of half dozen units: {ABC}, {D}, {E}, {F}, {G}, {H}, which can be arranged in 6! ways. Now, A, B and C inside their unit tin be bundled in 3! means, which gives full of vi!*3!=iv,320 dissimilar arrangements.
Answer: D.
Hope it'south clear.
quick question regarding this: Since the question doesn't specify which 3 people need to sit down together, how come up we don't take find ways of choosing 3 ppl out of the 8 to act as unit? Similar {DEF}, {AGF} etc etc.. I was thinking nosotros would have to do 8C3 ten 3! x six!... can you tell me where I am going wrong with my logic? Thanks for your help!
I'll try to give y'all my interpretation, waiting for some math skillful 
Choosing 3 people out of 8, you would summate all the possible sub-group of 3 people you could select from a group of 8. E.g., ABC, ABD, ABE, BCD, FGH, FBE, ... so on so forth.
The question specify "if 3 of them must sit together". Therefore it is non asking to find all the possible combinations in which 3 people can always sit next to each other. Paraphrasing, it is just asking "no affair who, consider that at that place are 3 people that decided they must always sit together (either ABC or ABD or ABE etc..): in this case, how many combinations can we create?" . That is why y'all accept to do your calculation only on i possible sub-group, non on all of them.
Hope it is clear...and correct!
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Re: In how many ways can y'all sit 8 people on a bench if 3 of [#permalink] 31 Oct 2017, 15:40
Gavan wrote:
In how many ways can y'all sit down eight people on a bench if 3 of them must sit together?
A. 720
B. two,160
C. 2,400
D. 4,320
East. forty,320
Since we are not given any names, nosotros tin can denote each person with a alphabetic character:
A, B, C, D, E, F, G, H
Permit's say A, B, and C must sit together; we care for [A-B-C] every bit a unmarried entity, and then nosotros have:
[A - B - C] - D - E - F - Yard - H
Nosotros run into that we have 6 total positions, which can exist bundled in vi! = 720 ways. We also can organize [A - B - C] in 3! = 6 ways.
And so, the full number of ways to adapt the group is 720 x 6 = 4,320 ways.
Answer: D
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Re: In how many means tin yous sit eight people on a bench if iii of [#permalink] 23 Nov 2019, 08:27
Gavan wrote:
In how many ways can you sit 8 people on a bench if iii of them must sit together?
A. 720
B. 2,160
C. 2,400
D. 4,320
E. twoscore,320
possible means = 3! for 3 people and 6! for total group of five single + 1 of 3 people
half dozen*6! = 4320
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Re: In how many ways can y'all sit down eight people on a bench if 3 of [#permalink] 23 Mar 2020, 15:59
Gavan wrote:
In how many means can you sit 8 people on a bench if 3 of them must sit together?
A. 720
B. 2,160
C. 2,400
D. 4,320
E. forty,320
8 people if three have to sit together can be written equally :
5+3= half dozen (considering three as one batch)
half-dozen people tin sit together in 6! ways and iii people (considered every bit 1 batch) can sit together in 3! ways.
total ways hence tin be 6!x3!= 4320
Answer: C
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Re: In how many means tin can you sit 8 people on a bench if 3 of [#permalink] 23 Mar 2020, 21:45
Take 3 persons equally a unit of measurement. And so total of 6 persons. Six people can sit in vi! Means. 3 people tin rearrange themselves in three! Ways. So 6! three! Ways. And so 720*6= 4320 means. Hope you lot understood.
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Re: In how many ways can you lot sit eight people on a demote if 3 of [#permalink] 05 Apr 2021, 03:57
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Re: In how many means can y'all sit down viii people on a bench if 3 of [#permalink]
05 Apr 2021, 03:57
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