Question 1208353
<pre>
We will first consider only 5 chairs and 5 people, and then think of how to
insert a vacant chair between two of the people appropriately.

[This is a round table problem.  I always complain about round table problems.
That's because in every one of them, we must pretend the totally unrealistic
situation that the table, chairs, and people are all placed on a huge turntable
that can be rotated in any direction without considering any rotation of the
huge turntable to be a different seating arrangement.  We know this is not the
way it is in reality, but we have to live with this anyway, because every math
book I've ever seen considers round table problems to be this unrealistic way.
So we'll assume it here.]

With that huge turntable in mind, there are only two possible configurations,
sex-wise, illustrated below, before the vacant chair is inserted. You may like
to contemplate whether there is any other possible configuration, but you'll
find that it will always be possible to rotate the huge turntable to show that
it's always and only one of these two. 
 
{{{drawing(50,50,-1.2,1.2,-1.45,.95,

locate(.5877852523,.8090169944,F),
locate(-.5877852523,.8090169944,F),
locate(.9510565163,-.3090169944,M),
locate(0,-1,M),
locate(-.9510565163,-.3090169944,M))}}}  or  {{{drawing(50,50,-1.2,1.2,-1.45,.95,

locate(.5877852523,.8090169944,M),
locate(-.5877852523,.8090169944,M),
locate(.9510565163,-.3090169944,F),
locate(0,-1,M),
locate(-.9510565163,-.3090169944,F))}}}  

</pre>1) The female students are adjacent.<pre> 
{{{drawing(50,50,-1.2,1.2,-1.45,.95,

locate(.5877852523,.8090169944,F),
locate(-.5877852523,.8090169944,F),
locate(.9510565163,-.3090169944,M),
locate(0,-1,M),
locate(-.9510565163,-.3090169944,M))}}}

It can only be like the configuration on the left above, but we may only 
insert the vacant chair in any of the 4 places that are NOT between the 
two females.

Arrange the females 2!=2 ways
Arrange the males 3!=6 ways
Insert the vacant chair any of 4 ways.
Answer: (2)(6)(4) = 48 ways.

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</pre>2) The female students are adjacent and the male students are adjacent.<pre>It can also only be like the configuration on the left above, except this
time the vacant chair can only be inserted in 2 ways, between a male and a
female. 

{{{drawing(50,50,-1.2,1.2,-1.45,.95,

locate(.5877852523,.8090169944,F),
locate(-.5877852523,.8090169944,F),
locate(.9510565163,-.3090169944,M),
locate(0,-1,M),
locate(-.9510565163,-.3090169944,M))}}}  

Arrange the females 2!=2 ways
Arrange the males 3!=6 ways
Insert the vacant chair either of 2 ways.
Answer: (2)(6)(2) = 24 ways.

--------------------------------------------------

</pre>3) No two female students are adjacent.<pre> 
It can be like either of the configurations above, and thus there are two cases.

{{{drawing(50,50,-1.2,1.2,-1.45,.95,

locate(.5877852523,.8090169944,F),
locate(-.5877852523,.8090169944,F),
locate(.9510565163,-.3090169944,M),
locate(0,-1,M),
locate(-.9510565163,-.3090169944,M))}}}  or {{{drawing(50,50,-1.2,1.2,-1.45,.95,

locate(.5877852523,.8090169944,M),
locate(-.5877852523,.8090169944,M),
locate(.9510565163,-.3090169944,F),
locate(0,-1,M),
locate(-.9510565163,-.3090169944,F))}}}

First case: It's like the configuration on the left, and the vacant chair can
only be inserted between the two females to keep them from being adjacent.

Arrange the females 2!=2 ways
Arrange the males 3!=6 ways
Insert the vacant chair only 1 way, between the two females
Answer: (2)(6)(1) = 12 ways.

Second case: It's like the configuration on the right, the vacant chair can
be placed between any two people, i.e., 5 places.

Arrange the females 2!=2 ways
Arrange the males 3!=6 ways
Insert the vacant chair any of 5 possible ways.
Answer: (2)(6)(5) = 60 ways.

Total for the two cases 12+60 = 72 ways.

Edwin</pre>