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sh3lsh

  • one year ago

How many positive integers less than 1000 have distinct digits? Is there a way to approach through combinatorics?

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  1. misty1212
    • one year ago
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    HI!!

  2. misty1212
    • one year ago
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    i would say they all have distinct values !

  3. sh3lsh
    • one year ago
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    Oh, I'm sorry. Have distinct digits!

  4. misty1212
    • one year ago
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    10 choices for the first digit, then 9 for the second and finally 8 for the third counting principle from there

  5. sepeario
    • one year ago
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    damn this could be hard..

  6. sepeario
    • one year ago
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    oh wait.

  7. sepeario
    • one year ago
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    1-digit integers: 1, ..., 9 - there are 9 integers. 2-digit integers: 10, ..., 99 - there are 90 integers, but in 9 of them (11, ..., 99) the two digits are the same. So there are 90 − 9 = 81 2-digit integers with distinct digits. Another way to get this answer is to consider the number of possibilities for each digit: the first digit can be any non-zero digit, so it has 9 choices. The second digit can be any digit except equal to the first one, so it has 9 choices too. There are 9 · 9 = 81 choices total. 3-digit integers: 100, ..., 999 - there are 900 integers total, but some of them have a repeating digit... The number of 3-digit integers with distinct digits can be counted as follows: the first digit can be any non-zero digit, so it has 9 choices. The second digit can be any digit except equal to the first one, so it has 9 choices too. Finally, the third digit can be any digit except equal to the first digit or the second digit, so it has 8 choices. There are 9 · 9 · 8 = 648 choices total. So there are 9 + 81 + 648 = 738 positive integers less than 1000 wit distinct digits.

  8. sepeario
    • one year ago
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    http://mail.csufresno.edu/~mnogin/math114fall04/41-18-sol.pdf

  9. sepeario
    • one year ago
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    or you could just read that ^ :)

  10. ybarrap
    • one year ago
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    You can use some combinatorics, but really just a little multiplication. 1000 does not have unique digits. It's out Let's look at the rest - 0 - 999 * 1-9 : 9 total, all distinct * 10-99 : 9*9=81 distinct, because 11,22,33,44,55,66,77,88,99 are removed (that is 90-9=81 distinct numbers). Note, in the MSD, 1-9 are allowed, in the second digit, 0-9 are allow, but we need to take out the digit that was used in MSD, leaving 9 digits for the LSD * 100-999: 9*9*8=648 distinct, for similar reasons Total = 9+81+648 = 738 distinct numbers

  11. sh3lsh
    • one year ago
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    Thanks a bunch you all!

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