document.write( "Question 1208345: \"In how many ways can the digits of the number 30348877 be arranged such that no two even digits are adjacent?\" \n" ); document.write( "

Algebra.Com's Answer #846708 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "In how many ways can the digits of the number 30348877 be arranged such that
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document.write( "We have four even digits  0, 4, 8, 8  and 4 odd digits  3, 3, 7, 7.\r\n" );
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document.write( "Two possible placements are  EOEOEOEO  or  OEOEOEOE,  where E is the placeholder \r\n" );
document.write( "for an even digit and O is the placeholder for an odd digit.\r\n" );
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document.write( "For odd digits with repetitions, as in the given number, there are\r\n" );
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document.write( "     =  = 6 different distinguishable permutations are possible \r\n" );
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document.write( "inside the group of odd numbers.\r\n" );
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document.write( "For even digits with repetitions, as in the given number, there are\r\n" );
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document.write( "     =  = 12 different distinguishable permutations are possible \r\n" );
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document.write( "inside the group of even numbers.\r\n" );
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document.write( "Of these 12 arrangements, the number of those that start with 0 is 3;  so these 3 should be subtracted from 12.\r\n" );
document.write( "Doing this way, we find that the number of arrangements of four even numbers in this problems, \r\n" );
document.write( "that do not start from 0 is 12-3 = 9.\r\n" );
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document.write( "Thus we have 6*12 = 72 arrangements of the type OEOEOEOE and 9*6 = 54 arrangements of the type EOEOEOEO.\r\n" );
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document.write( "The total is the sum 72 + 54 = 126.\r\n" );
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document.write( "Hence, in total, there are  126 different distinguishable arrangements as required in the problem.   ANSWER\r\n" );
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