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Homework Help: Sum of coeff.

  1. Mar 19, 2007 #1
    1. The problem statement, all variables and given/known data

    Evaluate:

    [tex]^{30}C_0 ^{30}C_{10}-^{30}C1 ^{30}C_11+...+^{30}C_{20} ^{30}C_{30}[/tex]

    Again, I know this is some coeff of the product of some series, but I dont know how to find the series or the coeff.


    I dont know how to go about it.
     
  2. jcsd
  3. Mar 19, 2007 #2

    JasonRox

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    Is this the whole question?

    To evaluate it, wouldn't you actually need to know what the [itex]C_i[/itex] actually are?

    Because if they're unknown constants, then I have no idea what is meant by evaluating that. Plus, your equation is not totally clear. You have a negative randomly put it where all the others are positives. You should explicitly write when the negatives are to be.
     
  4. Mar 19, 2007 #3

    HallsofIvy

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    Would those be the binomial coefficients? If so your nCi is more commonly written nCi.
     
  5. Mar 19, 2007 #4
    is the thing alternating or what?
     
  6. Mar 20, 2007 #5
    I'm assuming the thing is alternating..

    Try working out the first few terms...
    Alternately, try to write a formula for the nth term, and a formula for the (n+1)th term. See what happens when you add them together.
     
  7. Mar 20, 2007 #6
    Yes, those are binomial coefficients. The general term comes out to be

    [tex](-1)^n^{30}C_r ^{30}C_{10+r}[/tex] where r varies from 0 to 20.
     
  8. Mar 20, 2007 #7
    hint:

    1.[tex]\binom{a}{b}=\binom{a}{b-a}[/tex]

    2. look at
    [tex](1-x)^n(1+x)^n[/tex]

    in general, when you have two series
    [tex]S_1=\sum_n a_n x^n[/tex]
    [tex]S_2=\sum_n b_n x^n[/tex]

    [tex]S_1S_2=\sum_n\sum_{i+j=n} a_i b_jx^n[/tex]
     
    Last edited: Mar 20, 2007
  9. Mar 21, 2007 #8
    Can someone work the first two steps or something and I can try to work the rest out?
     
  10. Mar 23, 2007 #9
    reformulate the problem, basically you are asked to find,

    [tex]\sum_{i=0}^{20} (-1)^n\binom{30}{i}\binom{30}{20-i}[/tex]

    look at the function
    [tex]f(x)=\sum_{n}\left [\sum_{i=0}^{20} (-1)^n\binom{30}{i}\binom{30}{20-i}\right ]x^n[/tex]

    from the equation I posted last time
    [tex]S_1S_2=\sum_n\sum_{i+j=n}a_ib_jx^n=\sum_n\sum_{i=0}a_ib_{n-i}x^n[/tex]

    what can you conclude?
    what [itex]S_1[/itex] and [itex]S_2[/itex]
    should you construct to finish the problem?
     
    Last edited: Mar 23, 2007
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