Questions on Algebra: Combinatorics and Permutations answered by real tutors!

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Question 166109: How many different seating arrangements are possible for 6 people in 4 chairs?: How many different seating arrangements are possible for 6 people in 4 chairs?
Answer by jim_thompson5910(9928) About Me  (Show Source):
You can put this solution on YOUR website!

Since order does not matter (ie the seats have no unique positions), we must use the combination formula:


n!/(n-r)!r! Start with the combination formula.



6!/(6-4)!4! Plug in n=6 and r=4



6!/2!4! Subtract 6-4 to get 2


Expand 6!
(6*5*4*3*2*1)/2!4!


Expand 2!
(6*5*4*3*2*1)/(2*1)4!



(6*5*4*3*cross(2*1))/(cross(2*1))4! Cancel



(6*5*4*3)/4! Simplify


Expand 4!
(6*5*4*3)/(4*3*2*1)



360/(4*3*2*1) Multiply 6*5*4*3 to get 360



360/24 Multiply 4*3*2*1 to get 24



15 Now divide



So 6 choose 4 (where order doesn't matter) yields 15 unique combinations. So there are 15 possible ways that 6 people can sit in 4 chairs.