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Michigan State University
Raymond C.
Algebra
4 months, 1 week ago
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How many three-digit numbers can be formed using the digits $0,1,2,3,4,5,6,7,8,$ and $9 ?$ Repeated digits are allowed.
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Video Transcript
So we want to know how many three digit numbers can be formed using the digits 01 all the way up to nine. So there are 10 of these if repeats are allowed, which means this is not a permutation because in a permutation they would not be allowed. But let's put three little slots here. And if you want a three digit number now the question will be, can it start with zero? If it starts with zero and it's like a combination then zero could be in that spot. Let's let's assume that this must be one or higher. Then there would be nine ways to place a number here and then there would be any of the 10 digits here and any of the 10 digits here. So that would be 900. If your digit must be a non zero number. Now on the other hand, if you are allowed to have any digit here, Then it's going to be 10 times 10 times 10 or 1000. So it's going to depend on how you're going to define that first number. So it's either 900 or 1000 again, depending on can you play zero in that front position?
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Every three-digit number has digit 3 in it but doesn't have digit 0_ How many such kind of numbers are there with different digits?
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How many three digit numbers can be made from the digits 1,ldots,9 if repetitions of digits are not allowed?
Solution:
The proper approach to this question would be narrowing down the digits to sets that will always give numbers that are multiples of three.
{1,2,6}.sum() = 9
{2,2,5}.sum() = 9
{1,5,6}.sum() = 12
{2,4,6}.sum() = 12
{4,5,6}.sum() = 15
These sets of digits will only produce numbers that are multiples of 3, 3 digits long and are formed with numbers inside of the set. Each set will produce 6 unique numbers.
Finally to remove the numbers that end in 4, one simply needs to consider the sets that have a 4..
{2,4,6}
{4,5,6}
These produce 6 numbers each, 12 in total, and 4 of these numbers end in 4, namely: 264, 624, 564, 654.
This gives us a grand total of 6 * 3 + 8 = 26.
However, some of these numbers have duplicates. The {2,2,5} set creates 3 duplicate numbers.
225
252
225
252
525
525
Thus we must take away 3 from the final answer of 26, there are 23 numbers in total.
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How many 3-digit numbers are multiples of 3?
The list of multiples of 3 starts with 3, 6, 9, and goes in intervals of 3 forever. On this list, we want all the multiples of 3 that have exactly three digits. Below is the list of all the 3-digit numbers that are multiples of 3: 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204, 207, 210, 213, 216, 219, 222, 225, 228, 231, 234, 237, 240, 243, 246, 249, 252, 255, 258, 261, 264, 267, 270, 273, 276, 279, 282, 285, 288, 291, 294, 297, 300, 303, 306, 309, 312, 315, 318, 321, 324, 327, 330, 333, 336, 339, 342, 345, 348, 351, 354, 357, 360, 363, 366, 369, 372, 375, 378, 381, 384, 387, 390, 393, 396, 399, 402, 405, 408, 411, 414, 417, 420, 423, 426, 429, 432, 435, 438, 441, 444, 447, 450, 453, 456, 459, 462, 465, 468, 471, 474, 477, 480, 483, 486, 489, 492, 495, 498, 501, 504, 507, 510, 513, 516, 519, 522, 525, 528, 531, 534, 537, 540, 543, 546, 549, 552, 555, 558, 561, 564, 567, 570, 573, 576, 579, 582, 585, 588, 591, 594, 597, 600, 603, 606, 609, 612, 615, 618, 621, 624, 627, 630, 633, 636, 639, 642, 645, 648, 651, 654, 657, 660, 663, 666, 669, 672, 675, 678, 681, 684, 687, 690, 693, 696, 699, 702, 705, 708, 711, 714, 717, 720, 723, 726, 729, 732, 735, 738, 741, 744, 747, 750, 753, 756, 759, 762, 765, 768, 771, 774, 777, 780, 783, 786, 789, 792, 795, 798, 801, 804, 807, 810, 813, 816, 819, 822, 825, 828, 831, 834, 837, 840, 843, 846, 849, 852, 855, 858, 861, 864, 867, 870, 873, 876, 879, 882, 885, 888, 891, 894, 897, 900, 903, 906, 909, 912, 915, 918, 921, 924, 927, 930, 933, 936, 939, 942, 945, 948, 951, 954, 957, 960, 963, 966, 969, 972, 975, 978, 981, 984, 987, 990, 993, 996, 999 To get the answer to "How many 3-digit numbers are multiples of 3?" you simply need to count all the numbers above. Don't worry! You don't have to count them because we did it for you. There are 300 three digit (3-digit) numbers that are multiples of 3. Bonus: You may also be interested in the answer to: How many 3 digit counting numbers are not multiples of 3? There are 600 3-digit numbers that are not multiples of 3.3-digit Multiples Counter
Now you know how many 3-digit numbers are multiples of 3. Here you can get all the 3-digit numbers that are multiples of a different number.How many 3-digit numbers are multiples of 4?
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