Sorry, but the answer cannot be 420.  It is either 18 or 750.
They are ALL listed below.

If "4 even digits number" means "4-digit numbers with only even digits",
the answer is 18:

Choose the first digit 3 ways, as 2,4, or 6  
(1st digit can't be chosen as 0.)
Choose the second digit 3 ways as any of the 3 remaining digits.
Choose the third digit either of the 2 remaining digits.
Choose the fourth digit as the only remaining even digit.

3*3*2*1 = 18

Here are the only 18 4-digit numbers with only even digits:

2046, 2064, 2406, 2460, 2604, 2640, 
4026, 4062, 4206, 4260, 4602, 4620, 
6024, 6042, 6204, 6240, 6402, 6420.

--------------------------------------------

If "4 even digits number" means "4-digit even numbers",
the answer is 720:

Case 1.  The number has last digit 0:

Choose the last (fourth) digit 1 way, (as 0).
Choose the first digit 7 ways, (as 1,2,3,4,5,6,7).
Choose the second digit any of the 6 remaining ways.
Choose the third digit any of the 5 remaining ways.

That's 1*7*6*5 = 210 ways

Case 2.  The number has last digit 2,4, or 6:

Choose the fourth (last) digit 3 ways, (as 2, 4, or 6).
Choose the first digit any of the 5 remaining ways 
 other than 0 (can't choose 0 for the first digit)
Choose the second digit any of the 5 remaining ways.
Choose the third digit any of the 4 remaining ways.

That's 3*6*6*5 = 540 ways.

Total for both cases = 210+540 = 750

Here are all 750 4-digit even numbers, 30 rows with 25 in each row:


1024, 1026, 1032, 1034, 1036, 1042, 1046, 1052, 1054, 1056, 1062, 1064, 1072, 1074, 1076, 1204, 1206, 1230, 1234, 1236, 1240, 1246, 1250, 1254, 1256
1260, 1264, 1270, 1274, 1276, 1302, 1304, 1306, 1320, 1324, 1326, 1340, 1342, 1346, 1350, 1352, 1354, 1356, 1360, 1362, 1364, 1370, 1372, 1374, 1376
1402, 1406, 1420, 1426, 1430, 1432, 1436, 1450, 1452, 1456, 1460, 1462, 1470, 1472, 1476, 1502, 1504, 1506, 1520, 1524, 1526, 1530, 1532, 1534, 1536
1540, 1542, 1546, 1560, 1562, 1564, 1570, 1572, 1574, 1576, 1602, 1604, 1620, 1624, 1630, 1632, 1634, 1640, 1642, 1650, 1652, 1654, 1670, 1672, 1674
1702, 1704, 1706, 1720, 1724, 1726, 1730, 1732, 1734, 1736, 1740, 1742, 1746, 1750, 1752, 1754, 1756, 1760, 1762, 1764, 2014, 2016, 2034, 2036, 2046
2054, 2056, 2064, 2074, 2076, 2104, 2106, 2130, 2134, 2136, 2140, 2146, 2150, 2154, 2156, 2160, 2164, 2170, 2174, 2176, 2304, 2306, 2310, 2314, 2316
2340, 2346, 2350, 2354, 2356, 2360, 2364, 2370, 2374, 2376, 2406, 2410, 2416, 2430, 2436, 2450, 2456, 2460, 2470, 2476, 2504, 2506, 2510, 2514, 2516
2530, 2534, 2536, 2540, 2546, 2560, 2564, 2570, 2574, 2576, 2604, 2610, 2614, 2630, 2634, 2640, 2650, 2654, 2670, 2674, 2704, 2706, 2710, 2714, 2716
2730, 2734, 2736, 2740, 2746, 2750, 2754, 2756, 2760, 2764, 3012, 3014, 3016, 3024, 3026, 3042, 3046, 3052, 3054, 3056, 3062, 3064, 3072, 3074, 3076
3102, 3104, 3106, 3120, 3124, 3126, 3140, 3142, 3146, 3150, 3152, 3154, 3156, 3160, 3162, 3164, 3170, 3172, 3174, 3176, 3204, 3206, 3210, 3214, 3216
3240, 3246, 3250, 3254, 3256, 3260, 3264, 3270, 3274, 3276, 3402, 3406, 3410, 3412, 3416, 3420, 3426, 3450, 3452, 3456, 3460, 3462, 3470, 3472, 3476
3502, 3504, 3506, 3510, 3512, 3514, 3516, 3520, 3524, 3526, 3540, 3542, 3546, 3560, 3562, 3564, 3570, 3572, 3574, 3576, 3602, 3604, 3610, 3612, 3614
3620, 3624, 3640, 3642, 3650, 3652, 3654, 3670, 3672, 3674, 3702, 3704, 3706, 3710, 3712, 3714, 3716, 3720, 3724, 3726, 3740, 3742, 3746, 3750, 3752
3754, 3756, 3760, 3762, 3764, 4012, 4016, 4026, 4032, 4036, 4052, 4056, 4062, 4072, 4076, 4102, 4106, 4120, 4126, 4130, 4132, 4136, 4150, 4152, 4156
4160, 4162, 4170, 4172, 4176, 4206, 4210, 4216, 4230, 4236, 4250, 4256, 4260, 4270, 4276, 4302, 4306, 4310, 4312, 4316, 4320, 4326, 4350, 4352, 4356
4360, 4362, 4370, 4372, 4376, 4502, 4506, 4510, 4512, 4516, 4520, 4526, 4530, 4532, 4536, 4560, 4562, 4570, 4572, 4576, 4602, 4610, 4612, 4620, 4630
4632, 4650, 4652, 4670, 4672, 4702, 4706, 4710, 4712, 4716, 4720, 4726, 4730, 4732, 4736, 4750, 4752, 4756, 4760, 4762, 5012, 5014, 5016, 5024, 5026
5032, 5034, 5036, 5042, 5046, 5062, 5064, 5072, 5074, 5076, 5102, 5104, 5106, 5120, 5124, 5126, 5130, 5132, 5134, 5136, 5140, 5142, 5146, 5160, 5162
5164, 5170, 5172, 5174, 5176, 5204, 5206, 5210, 5214, 5216, 5230, 5234, 5236, 5240, 5246, 5260, 5264, 5270, 5274, 5276, 5302, 5304, 5306, 5310, 5312
5314, 5316, 5320, 5324, 5326, 5340, 5342, 5346, 5360, 5362, 5364, 5370, 5372, 5374, 5376, 5402, 5406, 5410, 5412, 5416, 5420, 5426, 5430, 5432, 5436
5460, 5462, 5470, 5472, 5476, 5602, 5604, 5610, 5612, 5614, 5620, 5624, 5630, 5632, 5634, 5640, 5642, 5670, 5672, 5674, 5702, 5704, 5706, 5710, 5712
5714, 5716, 5720, 5724, 5726, 5730, 5732, 5734, 5736, 5740, 5742, 5746, 5760, 5762, 5764, 6012, 6014, 6024, 6032, 6034, 6042, 6052, 6054, 6072, 6074
6102, 6104, 6120, 6124, 6130, 6132, 6134, 6140, 6142, 6150, 6152, 6154, 6170, 6172, 6174, 6204, 6210, 6214, 6230, 6234, 6240, 6250, 6254, 6270, 6274
6302, 6304, 6310, 6312, 6314, 6320, 6324, 6340, 6342, 6350, 6352, 6354, 6370, 6372, 6374, 6402, 6410, 6412, 6420, 6430, 6432, 6450, 6452, 6470, 6472
6502, 6504, 6510, 6512, 6514, 6520, 6524, 6530, 6532, 6534, 6540, 6542, 6570, 6572, 6574, 6702, 6704, 6710, 6712, 6714, 6720, 6724, 6730, 6732, 6734
6740, 6742, 6750, 6752, 6754, 7012, 7014, 7016, 7024, 7026, 7032, 7034, 7036, 7042, 7046, 7052, 7054, 7056, 7062, 7064, 7102, 7104, 7106, 7120, 7124
7126, 7130, 7132, 7134, 7136, 7140, 7142, 7146, 7150, 7152, 7154, 7156, 7160, 7162, 7164, 7204, 7206, 7210, 7214, 7216, 7230, 7234, 7236, 7240, 7246
7250, 7254, 7256, 7260, 7264, 7302, 7304, 7306, 7310, 7312, 7314, 7316, 7320, 7324, 7326, 7340, 7342, 7346, 7350, 7352, 7354, 7356, 7360, 7362, 7364
7402, 7406, 7410, 7412, 7416, 7420, 7426, 7430, 7432, 7436, 7450, 7452, 7456, 7460, 7462, 7502, 7504, 7506, 7510, 7512, 7514, 7516, 7520, 7524, 7526
7530, 7532, 7534, 7536, 7540, 7542, 7546, 7560, 7562, 7564, 7602, 7604, 7610, 7612, 7614, 7620, 7624, 7630, 7632, 7634, 7640, 7642, 7650, 7652, 7654


Be sure to tell your teacher that 420 is wrong, 
and that the correct answer is 750. 
Edwin