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A lecture hall contains 20 chairs, all lined in a row. What is the number of ways that five chairs can be chosen, so that no two chairs are adjacent?

 Aug 7, 2020
 #1
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By stars and bars, the number of ways is C(15,4) = 1365.

 Aug 7, 2020
 #2
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A lecture hall contains 20 chairs, all lined in a row. What is the number of ways that five chairs can be chosen, so that no two chairs are adjacent?

 

There will be 4 children each with an empty chair on their right, that is 8 seats taken.   But since they are tied in pairs, it is just 4 positions gone.

The fifth child will sit to the very right of all the others but there may not be an empty chair to his right.  So now there are 9 seats but just 5 positions taken.

There are now 11 seats left that all have their own position.

 

So there are 16 positions in total and I want to decide where the 11 can go.

 So that is 16C11 = 16C5 = 4368 

 

so I think there re 4368 arrangements and that is assuming that all the chairs are identical and all the children are identical.

 Aug 7, 2020

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