document.write( "Question 1174604: In how many ways can 4 boys and 4 girls be seated in a row if a. they can seat anywhere b. the boys and the girls are to be seated alternately? \n" ); document.write( "
Algebra.Com's Answer #800052 by mccravyedwin(408)\"\" \"About 
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In how many ways can 4 boys and 4 girls be seated in a row if \r
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document.write( "There are 8 people.\r\n" );
document.write( "We can choose the person to sit in the 1st chair 8 ways.\r\n" );
document.write( "That leaves 7 people to seat and 7 vacant chairs.\r\n" );
document.write( "We can choose the person to sit in the 2nd chair 7 ways.\r\n" );
document.write( "That leaves 6 people to seat and 6 vacant chairs.\r\n" );
document.write( "We can choose the person to sit in the 3rd chair 6 ways.\r\n" );
document.write( "That leaves 5 people to seat and 5 vacant chairs.\r\n" );
document.write( "We can choose the person to sit in the 4th chair 5 ways.\r\n" );
document.write( "That leaves 4 people to seat and 4 vacant chairs.\r\n" );
document.write( "We can choose the person to sit in the 5th chair 4 ways.\r\n" );
document.write( "That leaves 3 people to seat and 3 vacant chairs.\r\n" );
document.write( "We can choose the person to sit in the 6th chair 3 ways.\r\n" );
document.write( "That leaves 2 people to seat and 2 vacant chairs.\r\n" );
document.write( "We can choose the person to sit in the 7th chair 2 ways.\r\n" );
document.write( "That leaves 1 person to seat and 1 vacant chair.\r\n" );
document.write( "We can only seat this person in the 8th chair.\r\n" );
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document.write( "Answer = 8∙7∙6∙5∙4∙3∙2∙1 = 8! = 40320\r\n" );
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the boys and the girls are to be seated alternately?
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document.write( "Case 1.  The boys sit in seats 1,3,5,7 and the girls sit in seats 2,4,6,8.\r\n" );
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document.write( "We can choose the boy to sit in the 1st chair 4 ways.\r\n" );
document.write( "That leaves 3 boys to seat and 3 vacant chairs for boys.\r\n" );
document.write( "We can choose the boy to sit in the 3rd chair 3 ways.\r\n" );
document.write( "That leaves 2 boys to seat and 2 vacant chairs for boys.\r\n" );
document.write( "We can choose the boy to sit in the 5th chair 2 ways.\r\n" );
document.write( "That leaves 1 boy to seat and 1 vacant chair.\r\n" );
document.write( "We can only seat this boy in the 7th chair.\r\n" );
document.write( "We can choose the girl to sit in the 2nd chair 4 ways.\r\n" );
document.write( "That leaves 3 girls to seat and 3 vacant chairs for girls.\r\n" );
document.write( "We can choose the girl to sit in the 4th chair 3 ways.\r\n" );
document.write( "That leaves 2 girls to seat and 2 vacant chairs for girls.\r\n" );
document.write( "We can choose the girl to sit in the 6th chair 2 ways.\r\n" );
document.write( "That leaves 1 girl to seat and 1 vacant chair.\r\n" );
document.write( "We can only seat this girl in the 8th chair.\r\n" );
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document.write( "That's 4∙3∙2∙1∙4∙3∙2∙1 = 4!4! for Case 1.\r\n" );
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document.write( "Case 2.  The boys sit in seats 2,4,6,8 and the girls sit in seats 1,3,5,7.\r\n" );
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document.write( "That's the same answer as Case 1, 4!4!\r\n" );
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document.write( "Final answer:  4!4!+4!4! = 2(4!)2 =2(24)2 = 1152\r\n" );
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document.write( "Edwin
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