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Three friends and seven other people are randomly seated in a row. Specify an appropriate sample space to answer the following question. What is the probability that the three friends will sit next to each other?

The sample space has $10!$ equally probable results. My attempt was $$\frac{3!7!}{10!}$$ but it does not seem to be correct as it is $$\frac{8 \cdot 3!7!}{10!}$$ so why does $8$ figure in that expression?

N. F. Taussig
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Jim
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  • and your thoughts are ? –  Oct 08 '19 at 17:02
  • Welcome to MSE. You'll get a lot more help, and fewer votes to close, if you show that you have made a real effort to solve the problem yourself. What are your thoughts? What have you tried? How far did you get? Where are you stuck? This question is likely to be closed if you don't add more context. Please respond by editing the question body. Many people browsing questions will vote to close without reading the comments. – saulspatz Oct 08 '19 at 17:03
  • "The following two questions..." What is the second question? I only see one. – JMoravitz Oct 08 '19 at 17:03
  • I suggest you read my answer here talking about a different problem but touching on the nuances of selecting a sample space. – JMoravitz Oct 08 '19 at 17:05

1 Answers1

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The question didn't ask you to calculate the probability of the event occurring, it only asked you what you thought an appropriate sample space would be for the problem.

I agree that the sample space of size $10!$ which details all of the $10!$ equally likely ways in which the ten people can be lined up is an appropriate one to use and one of the first that many people would think of.

As for actually calculating the probability, $3!7!$ would only count in this context with this sample space the number of ways in which the first three people were the friends arranged in some order and the following seven people were the others arranged in some order.

You missed counting arrangements where the friends appeared in the middle or at the end of the other people. To correct the count, break into cases based on how many people were to the left of our group of friends. There are eight possibilities: $0$ people to the left, $1$ person to the left, etc... on up to $7$ people to the left of our group of friends.

JMoravitz
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