Find coefficient of $x^3y^4z^5$ in polynomial $(x + y + z)^8(x + y + z + 1)^8$
It is pretty easy to see that our goal is to choose from each multiplier in this polynomial $x,y,z,1$ in certain amounts. Since sum of powers equals $12$ and we have $16$ multipliers, we have to choose four $1$'s. There are $\displaystyle {8 \choose 4}$ ways to choose $1$ as part of our product (actually, four $1$'s). But how do we handle what has left? It seems like in such solution we have to consider many cases.
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