The square/cube/quartic... of a binomial/trinomial/multinomial?

Is there a general rule for finding $\displaystyle (a+b)^n/(a+b+c)^n/(a+b+c+...)^n$?

For all I know:

$\displaystyle (a+b)^1 = a^1+b^1 = a+b$

$\displaystyle (a+b)^2 = a^2+2ab+b^2$

$\displaystyle (a+b)^3 = a^3+3a^2b+3ab^2+b^3$

$\displaystyle [.................................................. .........]

$

$\displaystyle (a+b+c)^1 = a^1+b^1+c^1 = a+b+c$

$\displaystyle (a+b+c)^2 = (a+b+c)(a+b+c) = a(a+b+c)+c(a+b+c)+c(a+b+c) = $$\displaystyle a^2+ab+ac+ac+bc+c^2+ac+bc+c^2 = a^2+b^2+c^2+ab+ab+ac+ac+bc+bc =$ $\displaystyle a^2+b^2+c^2+2ab+2ac+2bc = a^2+b^2+c^2+2(ab+ac+bc)$

$\displaystyle [..........................................]$

But after that, it just gets tedious!