Problem - 4331
There are $10$ red, $10$ blue, and $10$ white balls. How many different ways are there to retrieve $16$ balls with all the three colors present.
The answer is the coefficient of the $x^{16}$ term in the expanded form of the following polynomial $$\begin{align*} f(x)=\ &\left(x+x^2+\cdots + x^{10}\right)^3\\ \\=\ &\frac{x^3(1-x^{10})^3}{(1-x)^3} \\=\ & x^3(1-3x^{10}+3x^{20}-x^{30})\sum_{k=0}^{\infty}\binom{2+k}{2}x^k \end{align*}$$
Therefore, the answer is $$\binom{2+13}{2}-3\cdot\binom{2+3}{2}=\boxed{75}$$