document.write( "Question 629274: Different positive four-digit integers are to be formed by using each of the digits 1,2,3,4 just once in each integer. How many different such integers can be formed if the digits 3 and 4 must NEVER be next to each other?\r
\n" ); document.write( "\n" ); document.write( "A. 4
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Algebra.Com's Answer #852476 by ikleyn(52943)\"\" \"About 
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\n" ); document.write( "Different positive four-digit integers are to be formed by using each of the digits 1,2,3,4 just once
\n" ); document.write( "in each integer. How many different such integers can be formed if the digits 3 and 4 must NEVER be next to each other?
\n" ); document.write( "A. 4
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\n" ); document.write( "C. 12
\n" ); document.write( "D. 16
\n" ); document.write( "E. 24
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\n" ); document.write( "\n" ); document.write( "        The idea is to calculate the number of all possible permutations of the four given digits\r
\n" ); document.write( "\n" ); document.write( "        and then subtract all unfavorable arrangements,  where  3  and  4  are next to each other.\r
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document.write( "The number of all possible permutations of the four given digits is 4! = 1*2*3*4 = 24.\r\n" );
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document.write( "The unfavorable arrangements are of two kinds.\r\n" );
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document.write( "    One kind of unfavorable arrangements is when adjacent 3 and 4 go in this order, '34'.\r\n" );
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document.write( "    Then we consider this block as one entity X, and we have 3 (three) objects to permutate:\r\n" );
document.write( "          the digits 1, 2 and the block X.  \r\n" );
document.write( "          It gives 3! = 1*2*3 = 6 four-digit numbers with the block '34'.\r\n" );
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document.write( "    The other kind of such arrangements is when adjacent 3 and 4 go in this order, '43'.\r\n" );
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document.write( "    Then we consider this block as one entity Y, and we have 3 objects to permutate:\r\n" );
document.write( "          the digits 1, 2 and the block Y.  \r\n" );
document.write( "          It gives 3! = 1*2*3 = 6 four-digit numbers with the block '43'.\r\n" );
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document.write( "Thus the number of favorable arrangements is  4! - 2*3! = 24 - 2*6 = 24 - 12 = 12.\r\n" );
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document.write( "ANSWER.  The number of favorable arrangements is 12.\r\n" );
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