<s>[INST] Find the coefficient of $x^{25}$ in $(1 + x^3 + x^8)^{10}$ 
Find the coefficient of $x^{25}$ in $(1 + x^3 + x^8)^{10}$.

Here is my solution I am looking to see if it is correct or if there is another way to do it, thanks!
The only way to form an $x^{25}$ term is to gather two $x^8$ and three $x^3$ . Since there are ${{10}\choose{2}} =45$ ways to choose two $x^8$ from the $10$ multiplicands and $8$ ways to choose three  ${{8}\choose{3}}= 56$ ways to choose $x^3$ from the remaining $8$multiplicands, the answer is using the product rule  $45×56 = 2520$.
 [/INST] Expressions like above can be evaluated by the multinomial Theorem:
$(a+b+c)^n = \sum_{k,l,m;k+l+m=n} \frac{n!}{k!l!m!}a^kb^lc^m$.
This sum means you sum over all possible $k,l,m$ with the condition that $k+l+m=n$.
Now you set $n=10,a=1,b=x^3,c=x^8$ and you obtain now that the $x^{25}$ Terms are obtained when $3l+8m=25$ for integer numbers $l,m$. This equation is satisfied only for $m=2,l=3$. Thus you obtain $k = 10-2-3=5$. It is $a^kb^lc^m = x^{25}$ and the resulting binomial coefficient is $\frac{10!}{5!2!3!} = 2520$. With above formula you obtain that the factor is $2520$ and that is correct!
</s>