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

#### buenogilabert

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

$\displaystyle [(1/6)^{10}]*[(10C2)*(14C5)-(10C1)*(20C11)+(26C17)]$

#### emakarov

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

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

$image=http://latex.codecogs.com/png.download?(1-s)^{10}=1+\binom{10}{1}s+\binom{11}{2}s^2+\dots+\binom{10+k-1}{k}s^k+\dots&hash=7c4463dadbce597b6a33afb92436cbc0$
(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

$\displaystyle \binom{10}{2}\binom{14}{5}-\binom{10}{1}\binom{20}{11}+\binom{26}{17}$

buenogilabert

#### buenogilabert

I spent 80 minutes with my tutor and he didn't know how to do it... Thanks a lot!

#### Plato

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