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Thread: Counting Question

  1. #1
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    Counting Question

    Alright guys, here's the question.

    How many five digit numbers with no beginning zeros (10000 - 99999) meet at least one of the following requirements?

    - Starts with a 3
    - Middle digit is 5
    - Last digit is 7


    Starts w/ 3:The first digit only has one way to be made, the remaining 4 each have 10 possibilities. Therefore 1 X 10^4
    Middle Dig 5::Because the first digit can't be 0, we have 9 ways to make the first digit, 10 ways for 2,4,5th digit, and one way for 3rd. Therefore 1 X 9 X 10^3
    Last Dig 7: Again we see first digit has 9 way, digits 2-4 each have 10 ways, and final digit has 1 way. Therefore 1 X 9 X 10^3.

    With this in mind, the number of numbers that meet at least one of the following is simple the number from each set added up?

    1 X 10^4 X 9^2 X 10^3 X 10^3

    Does this look correct?

    Thanks!
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  2. #2
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    Re: Counting Question

    first off you'd sum the 3 not multiply them

    secondly you have to worry about numbers that satisfy 2 or more of the criteria and only count them once
    Thanks from topsquark
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  3. #3
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    Re: Counting Question

    Meant to sum, hence the "added up" above the multiplication, lol.
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    Re: Counting Question

    Quote Originally Posted by PodoTheGreat View Post
    How many five digit numbers with no beginning zeros (10000 - 99999) meet at least one of the following requirements?
    - Starts with a 3
    - Middle digit is 5
    - Last digit is 7.
    Use the following notation $\mathcal{T}$ is the set of those numbers starting with three; $\mathcal{F}$ is the set of those numbers having middle digit five; $\mathcal{S}$ is the set of those numbers having last digit seven.
    The notation $\|\mathcal{A}\|$ is the number of terms in the set $\mathcal{A}$.

    This question is asking for $\|\mathcal{T\cup F\cup S}\|$ which means that you must know the inclusion/exclusion rules.

    $\|\mathcal{T\cup F\cup S}\|=\|\mathcal{T}\|+\|\mathcal{ F}\|+\|\mathcal{ S}\|-\|\mathcal{T\cap F}\|-\|\mathcal{T\cap S}\|-\|\mathcal{ F\cap S}\|+\|\mathcal{T\cap F\cap S}\|$

    Here is a start $\|\mathcal{F}\|=(9)(10)^3$ & $\|\mathcal{T\cap S}\|=(10)^3$
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  5. #5
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    Re: Counting Question

    Quote Originally Posted by romsek View Post
    first off you'd sum the 3 not multiply them
    secondly you have to worry about numbers that satisfy 2 or more of the criteria and only count them once
    See my reply.
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    Re: Counting Question

    Quote Originally Posted by romsek View Post
    secondly you have to worry about numbers that satisfy 2 or more of the criteria and only count them once
    So, the number of substrings that start w/ 3, mid dig 5, and last dig 7 = 1(10)1(10)1 = 10^2.
    Since that covers all strings that start w/ 3 and end with 7, start w/ 3 and middle digit 5, as well as middle digit 5 and end w/ 7, we simply subtract 10^2 from the final answer?
    Would that account for it?
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  7. #7
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    Re: Counting Question

    Quote Originally Posted by Plato View Post
    Use the following notation $\mathcal{T}$ is the set of those numbers starting with three; $\mathcal{F}$ is the set of those numbers having middle digit five; $\mathcal{S}$ is the set of those numbers having last digit seven.
    The notation $\|\mathcal{A}\|$ is the number of terms in the set $\mathcal{A}$.

    This question is asking for $\|\mathcal{T\cup F\cup S}\|$ which means that you must know the inclusion/exclusion rules.

    $\|\mathcal{T\cup F\cup S}\|=\|\mathcal{T}\|+\|\mathcal{ F}\|+\|\mathcal{ S}\|-\|\mathcal{T\cap F}\|-\|\mathcal{T\cap S}\|-\|\mathcal{ F\cap S}\|+\|\mathcal{T\cap F\cap S}\|$

    Here is a start $\|\mathcal{F}\|=(9)(10)^3$ & $\|\mathcal{T\cap S}\|=(10)^3$
    I can't quite follow what you're saying because of the formatting. Is that latex?
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  8. #8
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    Re: Counting Question

    Quote Originally Posted by PodoTheGreat View Post
    I can't quite follow what you're saying because of the formatting. Is that latex?
    It has nothing whatsoever with LaTeX.
    Do you know anything about inclusion/exclusion?
    If not, your trying to answer this question is a waste of time.
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  9. #9
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    Re: Counting Question

    100 meet the "3-5-7":

    1: 30507
    2: 30517
    ...
    99: 39587
    100: 39597
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  10. #10
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    Re: Counting Question

    Quote Originally Posted by PodoTheGreat View Post
    You're code didn't post correct and looked like a bunch of gibberish, not equations.
    What web-browser do you use. If you are not using a phone or a pad, but are using one of the most popular browsers you have reading the code.
    We have tested on many platforms.

    Here is a work-around. Click the reply with quote. Then remove all code tags: remove all [ tex][/tex][ /tex][/tex] tags & all $'s.
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