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

Oct 2011
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0
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:

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

Which is equal to

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

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

\(\displaystyle [(1/6)^{10}]*[(10C2)*(14C5)-(10C1)*(20C11)+(26C17)]\)
 
Oct 2009
5,577
2,017
\(\displaystyle (((1/6)*s)^{10})*((1-s^6)/(1-s))^{10}\)

Which is equal to

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

By the simplest form of the binomial theorem we have

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

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

(2)

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

\(\displaystyle \binom{10}{2}\binom{14}{5}-\binom{10}{1}\binom{20}{11}+\binom{26}{17}\)
 
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Oct 2011
5
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I spent 80 minutes with my tutor and he didn't know how to do it... Thanks a lot! :)
 
Aug 2006
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\(\displaystyle (((1/6)*s)^{10})*((1-s^6)/(1-s))^{10}\)
And the coefficient of \(\displaystyle s^{27}\)
Here is what I would do.
Because \(\displaystyle \left( {\frac{s}{6}} \right)^{10} = \frac{{s^{10} }}{{6^{10} }}\)

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

That is \(\displaystyle 1535040x^{17}\).

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.