In how many ways 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. 40,320
Is there any strategic approach to this question?
In how many ways can you sit 8 people on a bench if 3 of the
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This question could have been worded a little more clearly, but we'll take it to mean that there's enough room for all 8 people on the bench and that the 3 who wish to sit together does not change from scenario to scenario.ardz24 wrote:In how many ways 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. 40,320
Is there any strategic approach to this question?
Call the people A, B, C, D, E, F, G, H. Say A, B, and C insist on sitting together. If we fuse those people together, we now have A-B-C, D, E, F, G, H. In other words, we'll treat A-B-C as one entity. Along with A-B-C we have 5 more people to sit, so now we want to sit a total of six entities.. There are 6! or 720 ways we can do that.
But now we have to recognize that A-B-C don't have to be seated so that A is farthest to the left andC is farthest to the right. They could sit B-A-C or C-B-A, etc. In fact, there are 3!, or 6 ways we can sit A, B, and C.
720 * 6 = 4320. The answer is D
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Since we are not given any names, we can denote each person with a letter:ardz24 wrote:In how many ways 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. 40,320
A, B, C, D, E, F, G, H
Let's say A, B, and C must sit together; we treat [A-B-C] as a single entity, and so we have:
[A - B - C] - D - E - F - G - H
We see that we have 6 total positions, which can be arranged in 6! = 720 ways. We also can organize [A - B - C] in 3! = 6 ways.
So, the total number of ways to arrange the group is 720 x 6 = 4,320 ways.
Answer: D
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