The coefficient of $x^{42}$ in function $f(x)=(x+x^2+x^3+\cdots+x^{20})^3$ is the answer, which is $\boxed{190}$.

Let's say a, b, c are the values of your dice, with 1 <= a <= b <= c <=20. If a + b + c = 42, a + b + c <= 3c, that means 14 <= c. You can now take each value for c, {14, 15, 16, 17, 18, 19, 20} and see how many answers there are for a + b = 42 - c. For c = 14, you have a + b = 28, and the set of solutions is S = {(14, 14)} (remember that a <= b <= c), so |S| = 1 (note that you don't have any permutations) c = 15 , you have a + b = 27, S = {(14, 15)} (you have 3 permutations (14, 15, 15), (15, 14, 15), (15, 15, 14)) .... (and so on until c = 20) Then you add them up an should get the answer. Note: I'm not sure this is the solution you are looking for, it's a more time consuming one but it's logic and it will help you when out of ideas, also sorry if the format is bad, never wrote a aops comment in my life.

Inequality. wrote: The coefficient of $x^{42}$ in function $f(x)=(x+x^2+x^3+\cdots+x^{20})^3$ is the answer, which is $\boxed{190}$. Just wondering, what theorem is this? (And what chapter does it fall into or other things related to this)

folsky wrote: Inequality. wrote: The coefficient of $x^{42}$ in function $f(x)=(x+x^2+x^3+\cdots+x^{20})^3$ is the answer, which is $\boxed{190}$. Just wondering, what theorem is this? (And what chapter does it fall into or other things related to this) Generating Function