Coefficients Party, Part 4

The only 2 ways to get $x^2 \times y^2$ are by having $(term 1)^1 \times (term 2)^1$ or $(term 2)^2 \times (term 3)^2.$ However, The first case fails as you would need to multiply by $(term 3)^2$ as well as the powers have to add up to four. Hence, the second case is the only possible one.

What remains is to check in how many ways this combination will be achieved. Since we will have the product of 4 terms, 2 of which are $term 1,$ this number will be $C^4_2$ .

Hence, the required coefficient will be $(-3)^2 \times 6^2 \times C^4_2 = 1944.$

Finding the ways in which a term with powers of the variables $x^2y^2z^6$ can be obtained is rather like finding the way in which a physical quantity with particular unit "dimensions" can be formed from other physical quantities of specific units (the technique known as "dimensional analysis"). We have terms with "dimensions" $XY^2 \ , \ XZ \ , \$ and $YZ^2$ , so we are looking for the powers to which these terms are raised to produce the term in question.

Thus, we wish to determine $[ \ XY^2 \ ]^{\alpha} \cdot [ \ XZ \ ]^{\beta} \cdot [ \ YZ^2 \ ]^{\gamma} \ = \ X^2Y^2Z^6$ . Tracing what must be done for each variable gives us a system of equations

X : $\alpha + \beta \ = \ 2$ , Y : $2\alpha + \gamma \ = \ 2$ , Z : $\beta + 2\gamma \ = \ 6$ , subject to the overall requirement that the sum of the powers is $\alpha + \beta + \gamma \ = \ 4$ . This can be solved readily to tell us that $\ \alpha \ = \ 0 \ , \ \beta \ = \ 2 \ , \ \gamma \ = \ 2$ . [In this case, there is a unique solution; had this not been so, the dimensional analysis would have given us relations between the powers.]

Hence, there is only one set of terms in the total expansion that contains powers of the variables $x^2y^2z^6$ , which is $[ \ xy^2 \ ]^0 \cdot \ [ \ xz \ ]^2 \cdot [ \ yz^2 \ ]^2$ . The number of such terms is $\ \frac{4!}{0! 2! 2!} \ = \ 6$ , so their sum in the polynomial expansion is

$6 \ \cdot \ [ \ -3xz \ ]^2 \cdot [ \ 6yz^2 \ ]^2 \ = \ 6 \ \cdot \ 9 \ \cdot \ 36 \ x^2y^2z^6 \ = \ 1944 \ x^2y^2z^6$ . Hence the coefficient is 1944 .