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Math Help - No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

  1. #1
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    No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

    Find the number of non-negative integer solutions of the inequality
    x1 + x2 + x3 + x4 + x5 + x6 < 10.

    Answer:
    If this was -
    Find the number of non-negative integer solutions of the equation
    x1 + x2 + x3 + x4 + x5 + x6 = 10

    I would view this as having 10 identical items to distribute among 6 people and would use \binom{10+6-1}{10}.

    However Im not sure how to handle it with when it's an equality. Would I write it as a summation formula?

    \sum_{n=0}^9 {n+6-1\choose n}
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  2. #2
    Super Member TheChaz's Avatar
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    Re: No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

    "10 identical items..." - ??? What makes you think that they are identical?

    Take a look at this:
    number theory - Non-negative integral solutions of $X_1+X_2+X_3+X_4<n$ - Mathematics - Stack Exchange
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  3. #3
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    Re: No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

    They are identical in that they I am viewing this as having ten 'ones' that are being distributed among 6 people....
    Taking x1 + x2 + x3 + x4 + x5 + x6 = 10
    So x1 could have 5 'ones' and x2 - x6 could have one 'one' each. Which would be a solution for the equation.

    Am I going about this the wrong way with that summation formula in my first post?
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  4. #4
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    Re: No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

    Quote Originally Posted by nukenuts View Post
    Find the number of non-negative integer solutions of the inequality
    x1 + x2 + x3 + x4 + x5 + x6 < 10.
    Answer:
    If this was -
    Find the number of non-negative integer solutions of the equation
    x1 + x2 + x3 + x4 + x5 + x6 = 10
    I would view this as having 10 identical items to distribute among 6 people and would use \binom{10+6-1}{10}.
    However Im not sure how to handle it with when it's an equality. Would I write it as a summation formula?
    \color{blue}\sum_{n=0}^9 {n+6-1\choose n}
    Your solution is absolutely correct. Way to go.
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