document.write( "Question 1208353: how many ways can 3 male students and 2 female students be arranged in 6 chairs around a round table in the following cases:
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Algebra.Com's Answer #846721 by mccravyedwin(407)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We will first consider only 5 chairs and 5 people, and then think of how to\r\n" );
document.write( "insert a vacant chair between two of the people appropriately.\r\n" );
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document.write( "[This is a round table problem.  I always complain about round table problems.\r\n" );
document.write( "That's because in every one of them, we must pretend the totally unrealistic\r\n" );
document.write( "situation that the table, chairs, and people are all placed on a huge turntable\r\n" );
document.write( "that can be rotated in any direction without considering any rotation of the\r\n" );
document.write( "huge turntable to be a different seating arrangement.  We know this is not the\r\n" );
document.write( "way it is in reality, but we have to live with this anyway, because every math\r\n" );
document.write( "book I've ever seen considers round table problems to be this unrealistic way.\r\n" );
document.write( "So we'll assume it here.]\r\n" );
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document.write( "With that huge turntable in mind, there are only two possible configurations,\r\n" );
document.write( "sex-wise, illustrated below, before the vacant chair is inserted. You may like\r\n" );
document.write( "to contemplate whether there is any other possible configuration, but you'll\r\n" );
document.write( "find that it will always be possible to rotate the huge turntable to show that\r\n" );
document.write( "it's always and only one of these two. \r\n" );
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document.write( "  or    \r\n" );
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1) The female students are adjacent.
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document.write( "It can only be like the configuration on the left above, but we may only \r\n" );
document.write( "insert the vacant chair in any of the 4 places that are NOT between the \r\n" );
document.write( "two females.\r\n" );
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document.write( "Arrange the females 2!=2 ways\r\n" );
document.write( "Arrange the males 3!=6 ways\r\n" );
document.write( "Insert the vacant chair any of 4 ways.\r\n" );
document.write( "Answer: (2)(6)(4) = 48 ways.\r\n" );
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2) The female students are adjacent and the male students are adjacent.
It can also only be like the configuration on the left above, except this\r\n" );
document.write( "time the vacant chair can only be inserted in 2 ways, between a male and a\r\n" );
document.write( "female. \r\n" );
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document.write( "Arrange the females 2!=2 ways\r\n" );
document.write( "Arrange the males 3!=6 ways\r\n" );
document.write( "Insert the vacant chair either of 2 ways.\r\n" );
document.write( "Answer: (2)(6)(2) = 24 ways.\r\n" );
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3) No two female students are adjacent.
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document.write( "It can be like either of the configurations above, and thus there are two cases.\r\n" );
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document.write( "  or \r\n" );
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document.write( "First case: It's like the configuration on the left, and the vacant chair can\r\n" );
document.write( "only be inserted between the two females to keep them from being adjacent.\r\n" );
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document.write( "Arrange the females 2!=2 ways\r\n" );
document.write( "Arrange the males 3!=6 ways\r\n" );
document.write( "Insert the vacant chair only 1 way, between the two females\r\n" );
document.write( "Answer: (2)(6)(1) = 12 ways.\r\n" );
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document.write( "Second case: It's like the configuration on the right, the vacant chair can\r\n" );
document.write( "be placed between any two people, i.e., 5 places.\r\n" );
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document.write( "Arrange the females 2!=2 ways\r\n" );
document.write( "Arrange the males 3!=6 ways\r\n" );
document.write( "Insert the vacant chair any of 5 possible ways.\r\n" );
document.write( "Answer: (2)(6)(5) = 60 ways.\r\n" );
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document.write( "Total for the two cases 12+60 = 72 ways.\r\n" );
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document.write( "Edwin
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