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Math Help - Algebraic Solution

  1. #1
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    Algebraic Solution

    In an examination held by Sherwood Boarding School, the maximum marks for each of three papers is n and that for the fourth paper is 2n. Find the number of ways in which a candidate can get 3n marks.

    Ans. \frac{1}{6}(n+1)(5n2+10n+6)

    I'm stumped on this one. I tried hard but was unable to solve this algebraically . How should we approach this algebraically ? Request an algebraic solution.

    Thanks in advance !
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  2. #2
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    Re: Algebraic Solution

    Tough one !
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  3. #3
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    Re: Algebraic Solution

    How many solutions does the following system have

    a+b+c+d = 3n

    0\leq  a \leq  n

    0\leq  b \leq  n

    0\leq  c \leq  n

    0\leq  d \leq  2n

    where a, b, c, and d are integers? (Discrete Math / Combinatorics)
    Last edited by Idea; September 1st 2014 at 10:39 PM.
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    Re: Algebraic Solution

    Thanks Idea! That was cool. So it is 8n-1C5n-1. Am I right ?

    This comes out to be \frac{(8n-1)!}{3n!(5n-1)!}. But how do we solve from there ? Please advise .
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    Re: Algebraic Solution

    No, that's not right.

    Start with an easier question

    How many solutions does the following system have

    a+b+c+d=3n

    where a,b,c,d \geq  0?

    Answer: C(3n+3,3)=\frac{(3n+3)!}{(3n)!3!}
    Last edited by Idea; September 2nd 2014 at 09:57 AM.
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    Re: Algebraic Solution

    Hi Idea,

    I have tried to solve your expression \frac{(3n+3)!}{(3n)!3!} but I am not getting an answer \frac{1}{6}(n+1)(5n2+10n +6) . Maybe there is something missing in the equation. Please advise how to get to the solution and what should be the correct equation.
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  7. #7
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    Re: Algebraic Solution

    'my' expression is the answer to 'my' problem where a,b,c,d >= 0

    'your' expression is the answer to 'your' problem where 0 <= a,b,c <= n and 0 <= d <= 2n

    You can subtract some things from my answer to get to your answer

    What things? solutions for which a > n or b > n or c > n or d > 2n
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  8. #8
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    Re: Algebraic Solution

    Hi Idea,

    I have tried 3n+2C3 and 2n+3C3 but unable to get to the correct answer. Please elaborate how to form the correct equation and solve to reach the correct answer .
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  9. #9
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    Re: Algebraic Solution

    Try this

    C(3n+3,3)-3C(2n+2,3)-C(n+2,3)+3C(n+1,3)=

    \frac{1}{6} (n+1) \left(5 n^2+10 n+6\right)
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  10. #10
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    Re: Algebraic Solution

    That's really way over my head. I really can't relate that to a + b + c + d = 3n with the given constraints for a,b,c and d. Maybe I'm not ready for this yet. Can't help, haven't done any math beyond my 12th grade. Don't have any college math background.
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