Binomial Theorem IIII

1. Jul 26, 2013

reenmachine

1. The problem statement, all variables and given/known data

Use the binomial theorem to find the coefficient of $x^6y^3$ in $(3x-2y)^9$.

2. Relevant equations

$1+9+36+84+126+126+84+36+9+1$

(I used two lines for the lenght)

$1(3x)^9(-2y)^0+9(3x)^8(-2y)^1+36(3x)^7(-2y)^2+84(3x)^6(-2y)^3+126(3x)^5(-2y)^4$
$+126(3x)^4(-2y)^5+84(3x)^3(-2y)^6+36(3x)^2(-2y)^7+9(3x)^1(-2y)^8+1(3x)^0(-2y)^9$

(again using two lines because of the lenght)

$19683x^9 - 118098x^8y + 314928x^7y^2 -489888x^6y^3 +489888x^5y^4 -326592x^4y^5$
$+145152x^3y^6 - 41472x^2y^7 +6912xy^8 -512y^9$

So the coefficient of $x^6y^3$ would be $-489888$

any help will be greatly appreciated! thank you!

2. Jul 26, 2013

Staff: Mentor

You can check your result with WolframAlpha.
I don't see where you would need help.

3. Jul 26, 2013

reenmachine

thank you! I wasn't aware of that site :surprised

4. Jul 26, 2013

micromass

There's really no need to write out the full polynomial. You only need one term.

5. Jul 26, 2013

reenmachine

You're right , I guess since it was the first time I used the theorem without a perfect $(x+y)^n$ I wanted to verify it.

thank you!

6. Jul 27, 2013

HallsofIvy

The problem, according to you, said "Use the binomial theorem" and you did NOT do that.

The binomial theorem says that $(a+ b)^n= \sum_{i=0}^\infty \begin{pmatrix}n \\ i\end{pmatrix}a^ib^{n-i}$. Here a= 3x and b= -2y. You want "the coefficent of x6y3 with n= 9.

So this is the "i= 6" term: $\begin{pmatrix}9 \\ 6\end{pmatrix}(3x)^6(-2y)^3= \begin{pmatrix}9 \\ 6\end{pmatrix}(729)x^6(-8)y^3$ so the coefficient is $\begin{pmatrix}9 \\ 6\end{pmatrix}(729)(-8)= \begin{pmatrix}9 \\ 6\end{pmatrix}= 5832\begin{pmatrix}9 \\ 6\end{pmatrix}$.

Of course, $\begin{pmatrix}9 \\ 6 \end{pmatrix}= \frac{9!}{6!3!}= \frac{9(8)(7)}{6}= 3(4)(7)= 84$.

7. Jul 27, 2013

Theorem.

Yes, as HallsofIvy pointed out, the whole point here of using the Binomial Theorem is that you DO NOT actually have to do a full expansion. You just need to pick the appropriate terms out of the summation!

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