document.write( "Question 1207574: Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if all the children are next to each other, and all the adults are next to each other? (Two seatings are considered the same if one can be rotated to form the other.) \n" ); document.write( "

Algebra.Com's Answer #845534 by ikleyn(52780) $\"\"$ $\"About$

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\n" ); document.write( "Four children and four adults are to be seated at a circular table.
\n" ); document.write( "In how many different ways can they be seated if all the children are next to each other,
\n" ); document.write( "and all the adults are next to each other? (Two seatings are considered the same
\n" ); document.write( "if one can be rotated to form the other.)
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document.write( "Again, as in one of problems that I solved for you earlier today, we subdivide \r\n" );
document.write( "the people in two separate groups. One group is of 4 children; the other group \r\n" );
document.write( "is of 4 adults.\r\n" );
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document.write( "First, we consider all possible circular permutations of these two groups as \r\n" );
document.write( "of two whole objects.\r\n" );
document.write( "The number of all possible circular permutations of these two groups is\r\n" );
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document.write( "    (2-1)! = 1! = 1.\r\n" );
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document.write( "Next, there are 4! = 24 permutations inside the group of 4 children and \r\n" );
document.write( "another 4! = 24 permutations inside the group of 4 adults.\r\n" );
document.write( "These permutations are independent and produce 24*24 = 576 permutations,\r\n" );
document.write( "in total.\r\n" );
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document.write( "So, there are 576 circular permutations, in all.    ANSWER\r\n" );
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