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# Help!!

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In how many ways can 7 people be seated in a row of 9 chairs if Jane and Joe must sit next to each other?

Dec 16, 2018

#1
+109468
+2

In how many ways can 7 people be seated in a row of 9 chairs if Jane and Joe must sit next to each other?

Just consider that there are 8 chairs and 6 people.

2*  8P6

2  * 20160  =  40,320  ways.

I think that is right.

Dec 16, 2018
#2
+111357
+3

Note that Jane and Joe can occupy any of 8 positions

12

23

34

45

56

67

78

89

And they can be seated in 2 ways  for each of these

And note that, for each of these positions, we will have 7  other  seats  available for the other 5 people to sit  in

Call these other  empty seats A B C D E F G

So....we can assign any 5 of these for the  other people to  sit  in

And we can permute these in  P(7, 5) ways   = 2520 ways

So....the total ways =

8 * 2 * 2520   =  40,320  ways

Dec 16, 2018
#3
+109468
+1

Thanks Chris,

I probably should have explained that

8P6 is the number of ways 6 things can be chose from 8 when order matters.

I 'tied' the 2 friends together.  Which one was on the right and which on the left had to be accounted for, that is why i multiplied by 2.

Dec 16, 2018
#4
+111357
+2

Just tried this one for myself.....our answers agree...so....probably correct...!!!

CPhill  Dec 17, 2018