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Math Help - How to use binomial theorem to work out the coefficient of s^27

  1. #1
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    How to use binomial theorem to work out the coefficient of s^27

    Hi, I have the following problem that is solved, but I have no clue what formula the teacher uses to solve it... Thanks a lot for your help!

    I have:

    (((1/6)*s)^{10})*((1-s^6)/(1-s))^{10}

    Which is equal to

    ((1/6)^{10})*(1-10s^6+...)*(1+10s+...)

    And the coefficient of s^{27} (and the main thing I don't understand how the teacher gets it) is

    [(1/6)^{10}]*[(10C2)*(14C5)-(10C1)*(20C11)+(26C17)]
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  2. #2
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    Re: How to use binomial theorem to work out the coefficient of s^27

    Quote Originally Posted by buenogilabert View Post
    (((1/6)*s)^{10})*((1-s^6)/(1-s))^{10}

    Which is equal to

    ((1/6)^{10})*(1-10s^6+...)*(1+10s+...)
    The last line should be multiplied by s^{10}.

    By the simplest form of the binomial theorem we have

    (1-s^6)^{10}=1-\binom{10}{1}s^6+\binom{10}{2}s^{12}+\dots (1)

    Also, by the last formula in this Wiki section, we have

    (2)

    We have s^{10} that comes from the first factor ((1/6)*s)^{10}, so the rest of the expression must contribute s^{17} towards s^{27}. We can get s^{12} from (1) and s^{5} from (2); s^{6} from (1) and s^{11} from (2); and s^{0} from (1) and s^{17} from (2). The product of the corresponding coefficients make the three terms in the expression

    \binom{10}{2}\binom{14}{5}-\binom{10}{1}\binom{20}{11}+\binom{26}{17}
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  3. #3
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    Re: How to use binomial theorem to work out the coefficient of s^27

    I spent 80 minutes with my tutor and he didn't know how to do it... Thanks a lot!
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    Re: How to use binomial theorem to work out the coefficient of s^27

    Quote Originally Posted by buenogilabert View Post
    (((1/6)*s)^{10})*((1-s^6)/(1-s))^{10}
    And the coefficient of s^{27}
    Here is what I would do.
    Because \left( {\frac{s}{6}} \right)^{10}  = \frac{{s^{10} }}{{6^{10} }}

    we are looking for the coefficient of s^{17} in
    \left( {\frac{{1 - s^6 }}{{1 - s}}} \right)^{10}  = \left( {s^5  + s^4  + s^3  + s^2  + s + 1} \right)^{10} .

    That is 1535040x^{17}.

    I found that using a CAS. I don't know how much you know about generating polynomials. So this may not help you at all.
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