Sum of binomial coefficients with three variables

Question

What's the sum of coefficients of $(a+b+c)^8$?

Thanks in advance!

Answer 1

If you just imagine multiplying out the product using the distributive law, without collecting like terms, you end up with a bunch of terms like $acbbccab$ with $8$ factors each. Every 8-letter word built from the letters $a$, $b$, $c$ will appear exactly once, and there are $3^8$ such words.

When you then rearrange the sum by collecting like terms, the total sum of the coefficients will be exactly how many terms you had before you started collecting, that is, $3^8$. (This is because the coefficient of each of the terms is initially $1$).

Answer 2

That will be $3^8 = 6561$. In general, if $f(x,y,z) = (x + y + z)^n$, then the sum of the coefficients is $f(1,1,1)$ which is $3^n$

Answer 3

Taking the sum of coefficients just means setting all the variables equal to$~1$. So you've got $(1+1+1)^8=3^8=6561$.