Example 3: There are two red balls, one black ball, and one white ball. Questions: (1) How many different ways are there to select the balls? Try to enumerate them respectively; (2) If 3 balls are randomly selected each time, how many different ways are there to select them?
Let $x_{1}, x_{2}, x_{3}$ represent red, black, and white balls respectively. The ways to choose 2 red balls correspond to $\left(x_{1} t\right)^{0},\left(x_{1} t\right)^{1},\left(x_{1} t\right)^{2}$. Thus, the possible ways to choose red balls are represented by $1+x_{1} t+\left(x_{1} t\right)^{2}$, where 1 indicates not choosing any, $x_{1} t$ indicates choosing one red ball, and $\left(x_{1} t\right)^{2}$ indicates choosing two red balls. Similarly, $\left(1+x_{2} t\right)$ and $\left(1+x_{3} t\right)$ represent the possible ways to choose one black ball and one white ball, respectively. Therefore, the expansion of $\left(1+x_{1} t+\left(x_{1} t\right)^{2}\right)\left(1+x_{2} t\right)\left(1+x_{3} t\right)=1+\left(x_{1}+x_{2}+\right.$ $\left.x_{3}\right) t+\left(x_{1}^{2}+x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3}\right) t^{2}+\left(x_{1}^{2} x_{2}+x_{1}^{2} x_{3}+x_{1} x_{2} x_{3}\right) t^{3}+\left(x_{1}^{2} x_{2} x_{3}\right) t^{4}$ shows five scenarios. The number 1 represents the case of not choosing any ball, which has 1 way; choosing one ball has 3 ways, i.e., choosing one red, black, or white ball, $\cdots$, choosing four balls has 1 way: two red, one black, and one white.
Thus, the total number of different ways to choose the balls is $1+3+4+3+1=12$.