Question 218176
4 people 6 chairs
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sitting anywhere, the number is 15.
if the empty chairs have to be adjacent, the number is 5
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let's do adjacent first.
you have 6 chairs and 4 people.
that means 2 empty chairs.
if they have to be together, they become equivalent to 1 empty chair with the total number of chairs being 5.
you now have 5 chairs and one of them has to be empty. 
this can happen in 5 ways.
you get:
effff
fefff
ffeff
fffef
ffffe
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no other combination is possible.
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general formula for number of combinations is:
n! / ((x)!*(n-x)!)
formula for this problem is 5! / (1!*4!)
where:
1! represents x!
4! represents (n-x)!
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the first part of your problem is solved as follows:
you have 6 slots.
you can fill them with 4 people
that will leave 2 empty slots.
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the general formula for number of possible combinations of 4 out of 6 is once again n! / ((x)!*(n-x)!)
This becomes 6! / (2!*4!)
where:
n = 6
x = 2
n-x = 4
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The answer is 6*5*4*3*2*1 / 2*1*4*3*2*1  30/2 = 15
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there are 15 combinations possible.
They are:
eeffff
efefff
effeff
efffef
effffe
feefff
fefeff
feffef
fefffe
ffeeff
ffefef
ffeffe
fffeef
fffefe
ffffee
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You have 15 ways in which you can fill the 6 seats with 4 people assuming they can sit anywhere they want.
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You have 5 ways in which you can fill the 6 seats with 4 people assuming the empty seats have to be together.
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