This is a stars and bars problem. This is what the chapter is about.
Suppose we have exactly 10 dice. These are regular six-faced dice. We are now asked to determine the number of combinations where precisely one amount of dots/one particular face does not appear (so for instance not a 6, but we then ALWAYS want $1,2,3,4,5$ to be in there). Below is a photo
This arrangement is now allowed because it contains all numbers. Notice that we do not care about the order, but only about the amount of threes or fours for instance. We want that precisely one face is missing from a photo like this.
Examples of what is not allowed: 2 numbers are missing, 3 numbers are missing, etc.
Counting the combinations can be represented by the equation: $$x_1 + x_2 +x_3 + x_4 + x_ 5+ x_6 =10 $$ Where the 10 stands for the exact number of dice there are and the variable for the different "bins" they can be in. As we have six faces, we have six bins.
The problem that one outcome does not appear corresponds to one of the $x_i =0$, let us pick $x_6=0$ and then just multiply our final answer by $6$ because we could have chosen any of the six variables to be $0$. We also have the additional restriction that each of the remaining $x_j \geq 1$ and $i\neq j$. We introduce new variables $y_j=x_j -1$ and write: $$y_1 + y_2 +y_3 + y_4 + y_ 5 =5 $$ We now have to pick $5$ different variables and end up with a stars and bars problem, five different bins, 5 stars: $$f(k,n)= \binom{n+k-1}{k-1}$$ $$f(5,5)= \binom{5+5-1}{5-1}=\binom{9}{4}=126. $$
We multiply by $6$ to obtain all combinations and obtain $6 \cdot 126=756$.
Did I do this correctly?
Yes, the answer is correct, and how you arrived at it is a nice way as well.
The only thing one might want to emphasize, is that what you've calculated in the end is still a 1 to 1 mapping to the initial dice combinations that are possible, i.e.that "multiplying by 6" is correct, because if you choose different $x_i$ to be $0$, then the resulting sets of possible dice combinations are disjunct.