How many ways are there to get a sum of $25$ when $10$ distinct dice are rolled? I have come across the following problem and have given it a good attempt below. I am wondering if I have proceeded correctly, and if not if someone could show me the correct answer or maybe a more efficient solution, thanks!

How many ways are there to get a sum of $25$ when $10$ distinct dice are rolled?

Each die can be a number from $1$ to $6$. With $10$ dice, our generating function becomes:
$$g(x) = (x+x^2+x^3+x^4+x^5+x^6)^{10}$$
We want to find the coefficient of $x^{25}$. Observe that the expression inside the brackets is a finite geometric series with $a=x, r=x, n=6$. Thus we have 
$$(x+x^2+x^3+x^4+x^5+x^6)^{10}$$
$$=\left(\frac{x(1-x^6)}{(1-x)}\right)^{10}$$
$$=x^{10}(1-x^6)^{10}(1-x)^{-10}$$
Then,
$$(1-x^6)^{10}(1-x)^{-10}$$
$$=\sum \binom{10}{i}(-x^6)^i \cdot\sum\binom{-10}{j}(-x)^j$$
In order to get terms that involve $x^{15}$ there are $3$ combinations of $i$ and $j$ to consider so that $6i+j = 15$
Thus we have:
$$\left[ \binom{10}{0}(-1)^0\binom{-10}{15}(-1)^{15}\right] + \left[ \binom{10}{1}(-1)^1 \binom{-10}{9}(-1)^9\right] + \left[\binom{10}{2}(-1)^2\binom{-10}{3}(-1)^3\right]$$
$$ = \left[\binom{10}{0}\binom{-10}{15}(-1)\right]+\left[ \binom{10}{1} \binom{-10}{9} \right] + \left[ \binom{10}{2} \binom{-10}{3}(-1)\right] $$
$$=\left[ \binom{10}{0}\binom{10+15-1}{15}(-1)^{15}(-1)\right] + \left[ \binom{10}{1} \binom{10+9-1}{9}(-1)^9\right] + \left[ \binom{10}{2}\binom{10+3-1}{3}(-1)^3(-1)\right]$$
$$= \left[ \binom{10}{0}\binom{24}{15}\right] - \left[ \binom{10}{1}\binom{18}{9}\right] + \left[ \binom{10}{2}\binom{12}{3}\right]$$ 
$$= 831204$$

You will enjoy reading about Polynomial Coefficients here.  See specifically Theorem 2.1 in the referenced paper on the sum of discrete uniform random variables.
What you are looking for is what's referred to as multinomials.  Your approach tries to distill a multinomial into its basic binomial counter parts.  There is a recursive relationship between multinomials, as discussed in the referenced paper.
