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# Seating Arrangements

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There are 8 identical chairs around a circular table. In how many ways can 5 people be seated in 5 of these chairs if Richard wants to sit next to either Surya or Tamas (or both Surya and Tamas)?

(Consider two seating arrangements to be the same if one seating arrangement can be rotated to obtain the other.)

So one possible method is complementary counting but i can't really get the answer from that

Aug 9, 2022

#1
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Here's my attempt

Let's   let  the positions around the table look like this

R  ___  ____   _____ _____   ______   ______  ____

1     2      3          4         5           6             7            8

Richard sits in chair 1

Position  2 is just to the right of Richard and position 8  is just to his left

Case 1 -  Surya wants to sit on his right (in chair 1) but Tamas does not occupy the chair on his left

Tamas can choose any chair in the positions 3 - 7  and she will not sit next to Richard  =  5 choices

And the other 2 people can can occupy the remaining seats in  P(5,2)  =   20 ways

This gives us    5 *20 =   100 ways

Case 2 -  Surya  wants to sit on his left (in Chair 8)  but Tamas  does not occupy the chair to his right

Again, Tamas can shoose any chair in positions 3 -7  and she will agian not sit next to Richard = 5 choices

And again, the other two people can occupy  the remaining seats in P( 5,2)  = 20 ways

This again gives us 100 ways

So far, this = 200 ways

The cases where Tamas sits either to his right or left  but Surya does not sit next to him also  gives us 200 ways

So...we have  400 ways, so far

Case 3 -  both sit  next to him  (in chairs 2 and 8)

This can  be done in two  ways  and for each of these the other two people can occupy any two of the positions 3-7 in

P(5,2)   =20 ways

So this arrangement  =  2 * 20  = 40 ways

So....the total ways in  which either one (or both)  sit next to him =

200 (2)  +  40   =    440 ways

Aug 9, 2022
edited by CPhill  Aug 9, 2022