Determine the number of terms in simplified expression I've just started teaching myself algebra from a high school text-book and I stumbled upon this problem :

How many terms does the simplified form of $(a+b+c)^{2006}+(a-b-c)^{2006}$ have ? 

I found something on Wikipedia that appeared to be useful, the Multinomial Theorem, but It still led me to no result . Can somebody walk me through the solution ?
I'm much more interested in the thinking rather than the actual number.

You don't need the multinomial theorem right away. What you need to notice is that $(a+b+c)=a + (b + c)$ and $(a-b-c)=a-(b+c)$. This means that when you do the simple binomial expansion, say $x = a$ and $y = b+c$, you lose a bunch of terms. To be precise, you lose all the terms that would have an odd power of $y$. So now you are left with expanding the remaining $1004$ terms (you can check this number). The remaining terms get doubled. 
So now you have $\sum_{i=0}^{1003} 2 {2006 \choose 2i} x^{2006-2i}y^{2i}$. I'm not quite sure how to complete it from here but I'm sure some algebraic manipulation (with some wrestling) would so the trick!
