<s>[INST] How to solve $(z^1+z^2+z^3+z^4)^3$ using Pascals Triangle? In an exercise it seems I must use Pascal's triangle to solve this $(z^1+z^2+z^3+z^4)^3$. The result would be $z^3 + 3z^4 + 6z^5 + 10z^ 6 + 12z^ 7 + 12z^ 8 + 10z^ 9 + 6z^ {10} + 3z^ {11} + z^{12}$. But how do I use the triangle to get to that result? Personally I can only solve things like $(x+y)^2$ and $(x+y)^3$.
Thanks for any tips that may be given.
 [/INST] The expression factors as
$$z^3(1+z)^3(1+z^2)^3=z^3(1+3z+3z^2+z^3)(1+3z^2+3z^4+z^6).$$
Then I see no better way than to perform the multiply (though there is a symmetry)
$$\begin{align}
&1+3z+&3z^2+&z^3\\
&&3z^2+&9z^3+&9z^4+&3z^5\\
&&&&3z^4+&9z^5+&9z^6+&3z^7\\
&&&&&&z^6+&3z^7+&3z^8+z^9\\
\end{align}$$
</s>