Working through some combinatorics problems and am currently working on one involving a trinomial expansion of $$(x+y+z)^6$$ The question asks:
- How many terms are in this expansion?
- What is the coefficient for the term $xy^2z^3$?
- What is the sum of the coefficients of all the terms?
I have already solved parts $1$ and $2$ sufficiently however I am struggling a little on part $3$. Obviously, I could just brute force it and calculate $$\sum_{n_1,n_2,n_3} \frac{n!}{n_1!n_2!n_3!}$$ Where $n_i$ are the powers of the $3$ variables and $n_1 + n_2 + n_3 = n = 6$
I am hoping to get a little direction or hint as to how to approach this problem in a more systematic and slick way. Thanks!
Best Answer
For a polynomial $f$ in variables $X_1, X_2, \ldots, X_n$, if we write $$P = \sum_{i_1, \ldots, i_n} a_{i_1, \ldots, i_n} X_1^{i_1} \cdots X_n^{i_n},$$ then $$P(1, \ldots, 1) = \sum_{i_1, \ldots, i_n} a_{i_1, \ldots, i_n}. $$
In your case, the answer to $3.$ is $(1 + 1 + 1)^6 = 3^6$.
The sum that you showed at the end is also computables with a few tricks involving correctly re-arranging the sum in terms of $n_1$ then $n_2$. I guess it is also a good exercise.