document.write( "Question 1190087: A 4-digit number is selected from the numbers {1,2,3,4,5,6} where the digits are selected without replacement.
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Algebra.Com's Answer #821652 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "            The traditional and the standard formulation and the meaning of the problem is that the digits are used without repetition . . . \r
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\n" ); document.write( "\n" ); document.write( "In this problem, there are two restrictions:\r
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\n" ); document.write( "\n" ); document.write( "         (a)   the first (mostleft, thousands) digit must be 4, or 5, or 6;\r
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\n" ); document.write( "\n" ); document.write( "         (b)   the last digit (ones digit) must be 2, or 4, or 6 in order for the number be even.\r
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\n" ); document.write( "\n" ); document.write( "These restrictions are not independent; therefore, an accurate analysis is required.\r
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document.write( "Case 1.  Let the last digit be 2.\r\n" );
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document.write( "         Then the leading digit can be 4 or 5 or 6, giving 3 options;\r\n" );
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document.write( "         the second digit can be any of remaining 6-2 = 4 digits, giving 4 options;\r\n" );
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document.write( "         the third digit can be any of remaining 6-3 = 3 digits, giving 3 options.\r\n" );
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document.write( "         Thus in this case we have 3 * 4 * 3 = 36 possible 4-digit numbers.\r\n" );
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document.write( "Case 2.  Let the last digit be 4.\r\n" );
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document.write( "         Then the leading digit can be 5 or 6, giving 2 options;\r\n" );
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document.write( "         the second digit can be any of remaining 6-2 = 4 digits, giving 4 options;\r\n" );
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document.write( "         the third digit can be any of remaining 6-3 = 3 digits, giving 3 options.\r\n" );
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document.write( "         Thus in this case we have 2 * 4 * 3 = 24 possible 4-digit numbers.\r\n" );
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document.write( "Case 3.  Let the last digit be 6.     (In this case the analysis is similar to case 2).\r\n" );
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document.write( "         Then the leading digit can be 4 or 5, giving 2 options;\r\n" );
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document.write( "         the second digit can be any of remaining 6-2 = 4 digits, giving 4 options;\r\n" );
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document.write( "         the third digit can be any of remaining 6-3 = 3 digits, giving 3 options.\r\n" );
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document.write( "         Thus in this case we have 2 * 4 * 3 = 24 possible 4-digit numbers.\r\n" );
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document.write( "Thus the total of cases 1, 2 and 3  is  24 + 36 + 24 = 84 possible 4-digit numbers.\r\n" );
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document.write( "ANSWER.  There are 84 4-digit numbers under imposed conditions.\r\n" );
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