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Homework Help: Compute how many n-digit numbers

  1. Sep 3, 2013 #1
    1. The problem statement, all variables and given/known data
    Compute how many n-digit numbers can be made from the digits of at least one of {0,1,2,3,4,5,6,7,8,9 }
    Assume, repetition or order do not matter.

    2. Relevant equations
    ## a_{1}, a_{2}, ..., a_{n} ##

    3. The attempt at a solution
    10 choices for the 1st sub-index, 10 choices for the second sub-index, ..., 10 choices for the nth- sub-index.
    ## 10^{n} ## total possible combinations.
    I think that now we need to add 'n' for a set full of one identical digit. i.e.: {2,2,...,n-th}
    Now, n*9 for all possibilities

    Now, I need to take some n_i number into account in pairs and each of the n-2 numbers repeated for each number.
    So, groups of two repeated for each i-th number?
    Also, then I would extend this to include triple identical numbers and the rest (n-3) numbers in the set?
    And, so on...?

    I am sorry, if this does not make any sense or it is too messy.
    Could someone give me any guidance?
    Thank you.
  2. jcsd
  3. Sep 3, 2013 #2


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    How can order not matter for numbers?

    Is it OK to have a string of 0's for leading digits?

    I don't follow what you are doing in this last section. Is there something about the problem you haven't told us?
  4. Sep 4, 2013 #3
    The question is for a Theoretical Computer Science class.
    You might be right, but he said that it does not matter. My professor was very vague in posing the question. I asked it twice.
    He gave some examples:
    ## {0, 0, 0,0, ...,0_{n} }={0} ##
    ## {0, 0, 0, ..., 0, 1_{n}}={0,1}##

    Yes, it is.
    This counts as a valid number.
    ## { a_{0}, a_{1}, ..., a_{n} } \equiv {0_{0},0_{1},...,0_{n} } ##

    Sadly, I have told you everything that was given to me.

    In the last section, I wanted to start counting numbers as if repetition mattered, but I recalled that the professor said that they do not.

    I will for sure ask for more clarity next time, I see him.
  5. Sep 4, 2013 #4


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    OK. Since leading zeroes are OK it seems to me that your original calculation of ##10^n## should do it.
  6. Sep 4, 2013 #5
    Let me get back to you in a few days, I believe I can obtain more specific information about the problem. Thank you for your help so far.
  7. Sep 10, 2013 #6
    It turns out that order does matter after all. One 'gotta' love language.
    An example helped to elucidate.

    {0,1,2,3,...,8,9, ...<whatever>,...} is a vector of size n and 1 solution that satisfies the constraints.
    {3,4,7,...,9,2,1,...<whatever>,... } is another vector of size n and another solution.

    Let me work it out. If you have any leads, they are welcome.
    Thanks :>
  8. Sep 16, 2013 #7
    Please, check the solution in attachment.
    Apparently, it is incorrect. Can someone verify?
    I think that I am not taking into account cases such as {m, o ,m } or {1,0,1}, where there could be repetitions.

    The solution should be in the form:
    Order matters
    { ... , <at least digits from 0 to 9>,..., <any numbers>, ... }

    Thank you.

    Attached Files:

  9. Sep 22, 2013 #8
    So, here is the solution.

    ## \dfrac{ |n| !}{ |n_{1}|! |n_{2}|!... |n_{k}|!} \text{, where n is the size of the vector and the values in denominator are types of symbols in n that repeat.}## ##\text{For instance, if I have a vector called v={e, i, g, e, n, v, a, l, u, e}, there are say n1 type symbols.}## ##\text{n1 relates to, say, e and our |n1|=3. The numerator in the equation takes care of all combination including repetitions}## ##\text{and the denominator takes care of cases as the one just mentioned. } ##
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