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Math Help - How many decimal n-tuples contain at least one each of {1,2,3}?

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    How many decimal n-tuples contain at least one each of {1,2,3}?

    How many decimal n-tuples contain at least one each of {1,2,3}?
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    Quote Originally Posted by qtpipi View Post
    How many decimal n-tuples contain at least one each of {1,2,3}?
    There are a total of 10^{n} decimal n-tuples.
    How many of those contain none of {1,2,3}?
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    I'm not sure. Can you give another hint
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    Compute for the remainder

    Quote Originally Posted by qtpipi View Post
    I'm not sure. Can you give another hint
    Decimal n-tuples are n-tuples of \{1,2,3,4,6,7,8,9,0\} which has cardinality of 10.

    From PLATO's hint on computing the number of distinct n-tuples from a set of cardinality 10:

    Quote Originally Posted by Plato View Post
    There are a total of 10^{n} decimal n-tuples.
    How many of those contain none of 1,2,3?
    \{1,2,3,4,5,6,7,8,9,0\}\setminus\{1,2,3\}=\{4,5,6,  7,8,9,0\}

    Use PLATO's hint to compute how many n-tuples can be made from the remainder set \{4,5,6,7,8,9,0\} of cardinality 7. Then substract from the number of distinct n-tuples from a set of cardinality 10.
    Last edited by aleph1; April 23rd 2009 at 05:54 AM.
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    Quote Originally Posted by qtpipi View Post
    How many decimal n-tuples contain at least one each of {1,2,3}?
    qtpipi,

    Could you please clarify the question?

    Do you want the number of tuples that contain at least one each of all the digits 1,2,3-- for example, 93321-- or do you want the number of tuples that contain at least one 1, at least one 2, or at least one 3-- for example, 99111?
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    Quote Originally Posted by awkward View Post
    Could you please clarify the question?
    Do you want the number of tuples that contain at least one each of all the digits 1,2,3-- for example, 93321-- or do you want the number of tuples that contain at least one 1, at least one 2, or at least one 3-- for example, 99111?
    Surely there is nothing to be clarified!
    “At least one” is the complement of “none”.
    10^{10}-7^{10}.
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    Quote Originally Posted by Plato View Post
    Surely there is nothing to be clarified!
    “At least one” is the complement of “none”.
    10^{10}-7^{10}.
    There is, in my mind, a difference between "at least one" and "at least one each of (some set)".
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    Quote Originally Posted by awkward View Post
    There is, in my mind, a difference between "at least one" and "at least one each of (some set)".
    That is fair enough. I should have read the question more carefully.
    The number of decimals n-tuples not containing one of 1,2,3 is 3\cdot 9^n - 3\cdot 8^n + 7^n. WHY?
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    Quote Originally Posted by Plato View Post
    That is fair enough. I should have read the question more carefully.
    The number of decimals n-tuples not containing one of 1,2,3 is 3\cdot 9^n - 3\cdot 8^n + 7^n. WHY?
    Neat!

    I am sure there is more than one way to solve the problem. I would have said (looking at the original problem rather than the complementary question) that the number of decimal n-tuples which contain at least one each of 1, 2, and 3 is

    10^n - 3 \cdot 9^n + 3 \cdot 8^n - 7^n.

    Why? Because that is the coefficient of \frac{1}{n!} x^n in

    e^{7x}\;(e^x - 1)^3.



    (Exponential generating functions... gotta love them.)
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