document.write( "Question 984165: How many three-digit even numbers can be made with the digits 0,1,2,4,5,7, and 9 if a)repetitions are allowed b) no repetitions are allowed? \n" ); document.write( "

Algebra.Com's Answer #605097 by Edwin McCravy(20056) $\"\"$ $\"About$

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How many three-digit even numbers can be made with the digits
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document.write( "Choose the hundreds digit 6 ways 1,2,4,5,7,9\r\n" );
document.write( "Choose the tens digit 7 ways 0,1,2,4,5,7,9\r\n" );
document.write( "Choose the ones or units digit 3 ways 0,2,4\r\n" );
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document.write( "Solution (6)(7)(3) = 126 ways.\r\n" );
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document.write( "100, 102, 104, 110, 112, 114, 120, 122, 124, 140, \r\n" );
document.write( "142, 144, 150, 152, 154, 170, 172, 174, 190, 192, \r\n" );
document.write( "194, 200, 202, 204, 210, 212, 214, 220, 222, 224, \r\n" );
document.write( "240, 242, 244, 250, 252, 254, 270, 272, 274, 290, \r\n" );
document.write( "292, 294, 400, 402, 404, 410, 412, 414, 420, 422,\r\n" );
document.write( "424, 440, 442, 444, 450, 452, 454, 470, 472, 474, \r\n" );
document.write( "490, 492, 494, 500, 502, 504, 510, 512, 514, 520, \r\n" );
document.write( "522, 524, 540, 542, 544, 550, 552, 554, 570, 572, \r\n" );
document.write( "574, 590, 592, 594, 700, 702, 704, 710, 712, 714, \r\n" );
document.write( "720, 722, 724, 740, 742, 744, 750, 752, 754, 770, \r\n" );
document.write( "772, 774, 790, 792, 794, 900, 902, 904, 910, 912, \r\n" );
document.write( "914, 920, 922, 924, 940, 942, 944, 950, 952, 954, \r\n" );
document.write( "970, 972, 974, 990, 992, 994 \r\n" );
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document.write( "That's 12 rows of 10 and 6 on the bottom row.  That's 126 ways.\r\n" );
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\n" ); document.write( " b) no repetitions are allowed?
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document.write( "Case 1: 0 is the ones or units digit.\r\n" );
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document.write( "Choose the ones or units digit 1 way as 0\r\n" );
document.write( "Choose the hundreds digit 6 ways 1,2,4,5,7,9\r\n" );
document.write( "That leaves 5 unchosen digits that we can choose for the tens digit.\r\n" );
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document.write( "That's (1)(6)(5) = 30 ways\r\n" );
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document.write( "Case 2: 2 is the ones (or units) digit.\r\n" );
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document.write( "Choose the ones or units digit 1 way as 2\r\n" );
document.write( "Choose the hundreds digit 5 ways 1,4,5,7,9 (cannot choose 0)\r\n" );
document.write( "That leaves 5 unchosen digits that we can choose for the tens digit\r\n" );
document.write( "(can choose 0)\r\n" );
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document.write( "That's (1)(5)(5) = 25 ways\r\n" );
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document.write( "Case 3: 4 is the ones or units digit.\r\n" );
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document.write( "Same as case 2.\r\n" );
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document.write( "That's also 25 ways.\r\n" );
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document.write( "Grant total: 30+25+25 = 80 ways. \r\n" );
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document.write( "Checking by computer:\r\n" );
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document.write( "102, 104, 120, 124, 140, 142, 150, 152, 154, 170, \r\n" );
document.write( "172, 174, 190, 192, 194, 204, 210, 214, 240, 250, \r\n" );
document.write( "254, 270, 274, 290, 294, 402, 410, 412, 420, 450, \r\n" );
document.write( "452, 470, 472, 490, 492, 502, 504, 510, 512, 514, \r\n" );
document.write( "520, 524, 540, 542, 570, 572, 574, 590, 592, 594, \r\n" );
document.write( "702, 704, 710, 712, 714, 720, 724, 740, 742, 750, \r\n" );
document.write( "752, 754, 790, 792, 794, 902, 904, 910, 912, 914, \r\n" );
document.write( "920, 924, 940, 942, 950, 952, 954, 970, 972, 974\r\n" );
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document.write( "That's 8 rows of 10 each.  That's 80 ways.\r\n" );
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document.write( "Edwin

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