Melody

Thanks Catmg,

I have thought about it and it is deliciously simple. 

I will have to think on it some more though.

Thanks for encouraging me to continue thinking on it.

n the expansion of (ab + bc + ca)^{12}, find the coefficient of a^8 b^8 c^8.

$L[a(b+c)+bc]^{12}\\~\\ \displaystyle \sum_{r=0}^{12}\binom{12}{r}(bc)^{12-r}* [a(b+c)]^{r}\\~\\ \displaystyle \sum_{r=0}^{12}\binom{12}{r}(bc)^{12-r}* a^r(b+c)^{r}\\~\\ \displaystyle \sum_{r=0}^{12}\binom{12}{r}(bc)^{12-r}* a^r\left [\displaystyle \sum_{t=0}^r\binom{r}{t}b^rc^{t-r}\right]\\~\\ \displaystyle \sum_{r=0}^{12}\binom{12}{r}(bc)^{12-r}* a^r\left [\displaystyle \sum_{t=0}^r\binom{r}{t}b^tc^{r-t}\right]\\~\\ $

Now I am only interested in the terms where abc are all to the power of 8 so r=8

$\displaystyle \binom{12}{8}(bc)^{12-8}* a^8\left [\displaystyle \sum_{t=0}^8\binom{8}{t}b^tc^{8-t}\right]\\~\\ \displaystyle \binom{12}{8}b^4c^4* a^8\left [\displaystyle \sum_{t=0}^8\binom{8}{t}b^tc^{8-t}\right]\\~\\ So\;\; we\;\; need \;t=4\\~\\ \displaystyle \binom{12}{8}b^4c^4* a^8 \binom{8}{4}b^4c^{4}\\~\\ \displaystyle \binom{12}{8}\binom{8}{4} a^8 b^8c^{8}\\~\\$

The coefficiant will be  495*70 = 34650   

just as guest already told you.  

L[a(b+c)+bc]^{12}\\~\\
\displaystyle \sum_{r=0}^{12}\binom{12}{r}(bc)^{12-r}* [a(b+c)]^{r}\\~\\
\displaystyle \sum_{r=0}^{12}\binom{12}{r}(bc)^{12-r}* a^r(b+c)^{r}\\~\\
\displaystyle \sum_{r=0}^{12}\binom{12}{r}(bc)^{12-r}* a^r\left [\displaystyle \sum_{t=0}^r\binom{r}{t}b^rc^{t-r}\right]\\~\\
\displaystyle \sum_{r=0}^{12}\binom{12}{r}(bc)^{12-r}* a^r\left [\displaystyle \sum_{t=0}^r\binom{r}{t}b^tc^{r-t}\right]\\~\\

\displaystyle \binom{12}{8}(bc)^{12-8}* a^8\left [\displaystyle \sum_{t=0}^8\binom{8}{t}b^tc^{8-t}\right]\\~\\
\displaystyle \binom{12}{8}b^4c^4* a^8\left [\displaystyle \sum_{t=0}^8\binom{8}{t}b^tc^{8-t}\right]\\~\\
So\;\; we\;\; need \;t=4\\~\\
\displaystyle \binom{12}{8}b^4c^4* a^8 \binom{8}{4}b^4c^{4}\\~\\
\displaystyle \binom{12}{8}\binom{8}{4} a^8 b^8c^{8}\\~\\

Good answer Builderboi   

Good try Builderboi, but not quite right.

8/19 * 7/18 * 6/17 =

OR

8C3 /  19C3  =   

this is two different ways to do it, the answer is the same each time. 

So half the question was missing before.

You could have just admitted that.

The question doesn't make sense.  

Thanks Max,

I just spent time working through your solution by hand. I needed the practice.

I never would of got that first step without your guidence :)

Max wants to know that you learned something.

Not that you copied his final answer into a box and it came up magically with "correct"

Your 'thank you' is appreciated, 

Your 'that is correct' is insulting and suggest do me that you are just copying the final answer with no understanding at all.

Sorry, I have not answered it.

My post was intended for a different repeat question.

Thanks for catching it guest :)

Sorry, I put my 'answered' post on a different request by mistake. 

   Thereby causing double confusion.