5. Given that the generating function of a sequence is $f(t)=\left(t+t^{2}+t^{3}+\cdots\right)^{n}$, then the coefficient of $t^{\prime}$ in its expansion is $(\quad)$
A. $C$ :
B. $C_{r-1}^{n}$
C. $C_{r}^{n-1}$
5. D $\left(t+t^{2}+t^{3}+\cdots\right)^{n}=t^{n}(1-t)^{-n}=t^{n} \sum_{r=0}^{\infty} C_{n+r-1}^{r} t^{r}=\sum_{r=n}^{\infty} C_{r=n}^{r-\eta} t^{r}=\sum_{r=n}^{\infty} C_{r-1}^{n}-t^{r}$.

5. D $\left(t+t^{2}+t^{3}+\cdots\right)^{n}=t^{n}(1-t)^{-n}=t^{n} \sum_{r=0}^{\infty} \binom{n+r-1}{r} t^{r}=\sum_{r=n}^{\infty} \binom{r=n}{r-\eta} t^{r}=\sum_{r=n}^{\infty} \binom{r-1}{n} t^{r}$.