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Math Help - Set theory problem

  1. #1
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    Set theory problem

    Need help solving the following. I cannot find enough information in the text to begin solving the problem.

    1/4 of the five element subsets of (1,2,3...n) contain the element 7, determine n. n >5.
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  2. #2
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    Re: Set theory problem

    There are \begin{pmatrix}n \\ 5\end{pmatrix}= \frac{n!}{5! (n-5)!} subsets of {1, 2, 3, ..., n} that contain 5 numbers. 1/4 of that would be \frac{n!}{5!(n-5)! 4} and "1/4 of the five element subsets of (1,2,3...n) contain the element 7" implies that must be an integer.
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  3. #3
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    Re: Set theory problem

    I have been struggling with the equation for hours trying to solve for n without success. Is there enough information given to solve this problem?
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Set theory problem

    Yes, there is enough information given to solve for n.

    HallsofIvy has told you that 1/4 of the number of subsets containing 5 elements is:

    N=\frac{n!}{4\cdot5!(n-5)!}

    Now, to find the number of subsets of cardinality 5 containing 7 as an element, we may use the fundamental counting principle to state:

    N={n-1 \choose 4}=\frac{(n-1)!}{4!((n-1)-4)!}=\frac{(n-1)!}{4!(n-5)!}

    Hence, we have:

    \frac{n!}{4\cdot5!(n-5)!}=\frac{(n-1)!}{4!(n-5)!}

    Now, you just need to solve for n.
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  5. #5
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    Re: Set theory problem

    I think the equation I need to solve for n is

    1/4*(n,5)=(n-1,4)

    but I have no idea how to solve this equation
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  6. #6
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    Re: Set theory problem

    This equation is the same as the last equation in MarkFL2 post. Multiply both sides by (n-5)!. Then divide both sides by (n-1)! (remember that n!/(n-1)! = n). Then multiply oth sides by 4!.
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  7. #7
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    Re: Set theory problem

    Quote Originally Posted by wattskickin View Post
    I think the equation I need to solve for n is
    1/4*(n,5)=(n-1,4) but I have no idea how to solve this equation

    You have been told exactly what to do.

    Solve \frac{n!}{4\cdot5!(n-5)!}=\frac{(n-1)!}{4!(n-5)!}
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