SOLUTION: 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?

Choose the hundreds digit 6 ways 1,2,4,5,7,9
Choose the tens digit 7 ways 0,1,2,4,5,7,9
Choose the ones or units digit 3 ways 0,2,4

Solution (6)(7)(3) = 126 ways.

Checking by computer:

100, 102, 104, 110, 112, 114, 120, 122, 124, 140, 
142, 144, 150, 152, 154, 170, 172, 174, 190, 192, 
194, 200, 202, 204, 210, 212, 214, 220, 222, 224, 
240, 242, 244, 250, 252, 254, 270, 272, 274, 290, 
292, 294, 400, 402, 404, 410, 412, 414, 420, 422,
424, 440, 442, 444, 450, 452, 454, 470, 472, 474, 
490, 492, 494, 500, 502, 504, 510, 512, 514, 520, 
522, 524, 540, 542, 544, 550, 552, 554, 570, 572, 
574, 590, 592, 594, 700, 702, 704, 710, 712, 714, 
720, 722, 724, 740, 742, 744, 750, 752, 754, 770, 
772, 774, 790, 792, 794, 900, 902, 904, 910, 912, 
914, 920, 922, 924, 940, 942, 944, 950, 952, 954, 
970, 972, 974, 990, 992, 994 

That's 12 rows of 10 and 6 on the bottom row.  That's 126 ways.

Case 1: 0 is the ones or units digit.

Choose the ones or units digit 1 way as 0
Choose the hundreds digit 6 ways 1,2,4,5,7,9
That leaves 5 unchosen digits that we can choose for the tens digit.

That's (1)(6)(5) = 30 ways

Case 2: 2 is the ones (or units) digit.

Choose the ones or units digit 1 way as 2
Choose the hundreds digit 5 ways 1,4,5,7,9 (cannot choose 0)
That leaves 5 unchosen digits that we can choose for the tens digit
(can choose 0)

That's (1)(5)(5) = 25 ways

Case 3: 4 is the ones or units digit.

Same as case 2.

That's also 25 ways.

Grant total: 30+25+25 = 80 ways. 

Checking by computer:

102, 104, 120, 124, 140, 142, 150, 152, 154, 170, 
172, 174, 190, 192, 194, 204, 210, 214, 240, 250, 
254, 270, 274, 290, 294, 402, 410, 412, 420, 450, 
452, 470, 472, 490, 492, 502, 504, 510, 512, 514, 
520, 524, 540, 542, 570, 572, 574, 590, 592, 594, 
702, 704, 710, 712, 714, 720, 724, 740, 742, 750, 
752, 754, 790, 792, 794, 902, 904, 910, 912, 914, 
920, 924, 940, 942, 950, 952, 954, 970, 972, 974

That's 8 rows of 10 each.  That's 80 ways.

Edwin