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four married couple went to attend a concert in how many ways can they be arranged:-\r
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\r\n" ); document.write( "We can choose the couple to sit on the far left in 4 ways.\r\n" ); document.write( "We can choose the way they sit 2 ways. (husband left of wife or wife left of husband).\r\n" ); document.write( "\r\n" ); document.write( "We can choose the couple to sit next to the far left in 3 ways.\r\n" ); document.write( "We can choose the way they sit 2 ways. (husband left of wife or wife left of husband).\r\n" ); document.write( "\r\n" ); document.write( "We can choose the couple to sit next to the far right in 2 ways.\r\n" ); document.write( "We can choose the way they sit 2 ways. (husband left of wife or wife left of husband)\r\n" ); document.write( "\r\n" ); document.write( "We can choose the couple to sit on the far right in just 1 way.\r\n" ); document.write( "We can choose the way they sit 2 ways. (husband left of wife or wife left of husband).\r\n" ); document.write( "\r\n" ); document.write( "That's 4×2×3×2×2×2×1×2 = 4!×24 = 384 ways\r\n" ); document.write( "
\n" ); document.write( "(b)if there are no restrictions
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\r\n" ); document.write( "Choose the person to sit in the 1st seat 8 ways.\r\n" ); document.write( "Choose the person to sit in the 2nd seat 7 ways.\r\n" ); document.write( "Choose the person to sit in the 3rd seat 6 ways.\r\n" ); document.write( "Choose the person to sit in the 4th seat 5 ways.\r\n" ); document.write( "Choose the person to sit in the 5th seat 4 ways.\r\n" ); document.write( "Choose the person to sit in the 6th seat 3 ways.\r\n" ); document.write( "Choose the person to sit in the 7th seat 2 ways.\r\n" ); document.write( "Choose the person to sit in the 8th seat 1 ways.\r\n" ); document.write( "\r\n" ); document.write( "That's 8×7×6×5×4×3×2×1 = 8! = 40320 ways.\r\n" ); document.write( "\r\n" ); document.write( "---------------------------------------------------\r\n" ); document.write( "
\n" ); document.write( "(c)if each wife does not sit next to her husband
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document.write( "Wow! This is the only hard one of the 3.\r\n" );
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document.write( "It's 8! minus all the unwanted seating arrangements:\r\n" );
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document.write( "So we have to calculate the number of unwanted seating arrangements\r\n" );
document.write( "this way:\r\n" );
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document.write( "Let A = the unwanted case that couple 1 sits together.\r\n" );
document.write( "Let B = the unwanted case that couple 2 sits together.\r\n" );
document.write( "Let C = the unwanted case that couple 3 sits together.\r\n" );
document.write( "Let D = the unwanted case that couple 4 sits together.\r\n" );
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document.write( "So we want N(A or B or C or D)\r\n" );
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document.write( "The \"sieve\" formula for the answer (sometimes called the \"inclusion and\r\n" );
document.write( "exclusion\" formula is:\r\n" );
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document.write( "N(A or B or C or D) = N(A) + N(B) + N(C) + N(D)\r\n" );
document.write( " - N(A&B) - N(A&C) - N(A&D) - N(B&C) - N(B&D) - N(C&D)\r\n" );
document.write( " + N(A&B&C) + N(A&B&D) + N(A&C&D) + N(B&C&D)\r\n" );
document.write( " - N(A&B&C&D)\r\n" );
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document.write( "It's easy to see that:\r\n" );
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document.write( "N(A) = N(B) = N(C) = N(D)\r\n" );
document.write( "N(A&B) = N(A&C) = N(A&D) = N(B&C) = N(B&D) = N(C&D)\r\n" );
document.write( "N(A&B&C) = N(A&B&D) = N(A&C&D) = N(B&C&D)\r\n" );
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document.write( "So the \"sieve\" formula becomes\r\n" );
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document.write( "N(A or B or C or D) = 4N(A) - 6N(A*B) + 4N(A&B&C) - N(A&B&C&D)\r\n" );
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document.write( "-----------------------------\r\n" );
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document.write( "We calculate N(A)\r\n" );
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document.write( "We will use (x,y) to mean that a couple sits in seats #x and #y\r\n" );
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document.write( "Choose couple 1's seats 7 ways: (1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8)\r\n" );
document.write( "Choose the way they can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
document.write( "Seat the other 6 people 6! ways.\r\n" );
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document.write( "That's 7×2!×6! = 10080\r\n" );
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document.write( "N(A&B) = 10080\r\n" );
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document.write( "-----------------------\r\n" );
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document.write( "We calculate N(A&B)\r\n" );
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document.write( "Couples 1 and 2 can use 4 seats in these 15 ways:\r\n" );
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document.write( "{(1,2),(3,4)}, {(1,2),(4,5)}, {(1,2),(5,6)}, {(1,2),(6,7)}, {(1,2),(7,8)},\r\n" );
document.write( "{(2,3),(4,5)}, {(2,3),(5,6)}, {(2,3),(6,7)}, {(2,3),(7,8)}, {(3,4),(5,6)},\r\n" );
document.write( "{(3,4),(6,7)}, {(3,4),(7,8)}, {(4,5),(6,7)}, {(4,5),(7,8)}, {(5,6),(7,8)}\r\n" );
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document.write( "Choose whether couple 1 is left of couple 2 or vice-versa in 2! ways:\r\n" );
document.write( "Choose the way couple 1 can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
document.write( "Choose the way couple 2 can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
document.write( "Seat the remaining 4 people 4! ways.\r\n" );
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document.write( "That's 15×2!×2!×2!×4! = 2880 ways.\r\n" );
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document.write( "N(A&B) = 2880\r\n" );
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document.write( "---------------------------\r\n" );
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document.write( "We calculate N(A&B&C) \r\n" );
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document.write( "Couples 1, 2 and 3 can use 6 seats in these 10 ways:\r\n" );
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document.write( "{(1,2),(3,4),(5,6)}, {(1,2),(3,4),(6,7)}, {(1,2),(3,4),(7,8)},\r\n" );
document.write( "{(1,2),(4,5),(6,7)}, {(1,2),(4,5),(7,8)}, {(1,2),(5,6),(7,8)}, \r\n" );
document.write( "{(2,3),(4,5),(6,7)}, {(2,3),(4,5),(7,8)}, {(2,3),(5,6),(7,8)},\r\n" );
document.write( "{(3,4),(5,6),(7,8)}\r\n" );
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document.write( "We can choose the order of the 3 couples is 3! or 6 ways.\r\n" );
document.write( "Choose the way couple 1 can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
document.write( "Choose the way couple 2 can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
document.write( "Choose the way couple 3 can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
document.write( "Seat the remaining 2 people 2! ways.\r\n" );
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document.write( "That's 10×3!×2!×2!×2!×2! = 960 ways.\r\n" );
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document.write( "N(A&B&C) = 960\r\n" );
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document.write( "--------------------------------------\r\n" );
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document.write( "We calculate N(A&B&C&D)\r\n" );
document.write( "(They use all 8 seats)\r\n" );
document.write( "Choose the order of the 4 couples 4! = 24 ways\r\n" );
document.write( "Choose the way couple 1 can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
document.write( "Choose the way couple 2 can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
document.write( "Choose the way couple 3 can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
document.write( "Choose the way couple 4 can sit 2! ways: (husband,wife) or (wife,husband)\r\n" );
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document.write( "That's 4!×2!×2!×2!×2! = 384\r\n" );
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document.write( "----------------------------\r\n" );
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document.write( "Substitute in \"sieve\" formula to get the number of unwanted seating\r\n" );
document.write( "arrangements:\r\n" );
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document.write( "N(A or B or C or D) = 4N(A) - 6N(A&B) + 4N(A&B&C) - N(A&B&C&D)\r\n" );
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document.write( "N(A or B or C or D) = 4(10080) - 6(2880) + 4(960) - 384 = 26496\r\n" );
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document.write( "---------------------------\r\n" );
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document.write( "Subtract from 8!:\r\n" );
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document.write( "8! - 26496 = 40320 - 26496 = 13824.\r\n" );
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document.write( "Answer: 13824\r\n" );
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document.write( "Edwin \n" );
document.write( "