What is (2a+b-c)^8. Or at least what is the sum of the coefficients?

Question

anonymous · Answer

my guess is $4^8$ but  i could be wrong

tkhunny · Answer

Good guess, but not quite.

a = b = c = 1

(2a + b - c) = 2

$2^{8}$

This principle is nicely demonstrated with smaller exponents.

$(2a + b - c)^{1} = 2a + b - c = 2^{1}$

$(2a + b - c)^{2} = 4a^{2}+4ab-4ac+b^{2}-2bc+c^{2} = 2^{2} = 4$

Some very tedious induction would demonstrate it definitively.

anonymous · Answer

So this means that the answer would be$$2^{8}$$?

tkhunny · Answer

Didn't I say that already?

I also decided that the induction wouldn't be all that tedious.

anonymous · Answer

Just making sure. Sorry

tkhunny · Answer

No, no.  Didn't you see my big smile when I said that?  :-)

anonymous · Answer

Yes. :) It is simply late and I have been doing math for a while, and I only saw the bottom half of the post...