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There are 10 persons to be seated on chairs with numbers 1 to 10. The first person first comes and can seat on any chair. Then for i = 2, 3, . . . , 10, the i-th person enters and takes the seat i if it is available, otherwise any other seat. In how many ways can they be seated?

I tried to make different cases and tried easier related questions to get a good idea to approach this one. But I still can't find a good approach. I just want a hint of some kind to get started on the correct path.

user242803
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  • Related: Taking Seats on a Plane. The linked problem has the same setup however rather than asking for a count of arrangements instead asks for a probability that the last seat is occupied by the last person. – JMoravitz May 08 '20 at 13:59
  • As for actually counting this... as for a hint on what you might think about or use as an approach, what do you notice about the cyclic structure of any such arrangement of the people in chairs? Can you try counting the permutations who have that specific property? – JMoravitz May 08 '20 at 14:01

1 Answers1

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A recurrence relation can be established as follows.

If there are $n$ persons (in this question $n=10$) then number of such arrangements $P(n)=1+P(n-1)+P(n-2)+.....+P(1)$.

Because, if the first person chooses $k$th position then for that configuration there are $P(n-k+1)$ ways.

So, $P(n)=1+P(n-1)+P(n-2)+.....+P(1)=P(n-1)+[P(n-2)+P(n-3)+...+P(1)]=2P(n-1)=2^{(n-1)}P(1)=2^{(n-1)}$

Alapan Das
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  • I don't understand Why for every i'th person there are P(n-1) ways. Say for example, 1 person takes the 5'th seat now the fifth person only has 6 option to select as second to fourth seats are already occupied. So the ways available for i'th person should be dependent on i. – user242803 May 08 '20 at 15:20
  • Yes. You are right. Thank you. I thought something different. – Alapan Das May 08 '20 at 15:38
  • What I did earlier would count for some problem like" If the $1$st person chooses $i$th position, then all the persons after $i$ would take their sits, not the previous comers. – Alapan Das May 08 '20 at 15:50