We want to prove that:
$$\sum_{t=1}^{r} t\binom{n-t}{n-r}=\binom{n+1}{r-1}$$
I have used basic properties of binomial coefficients like $$\binom{n+1}{r+1}= \frac{n+1}{r+1}\binom{n}{r}$$ but they seems to be of no use.
I have managed to prove that :
$$\sum_{t=0}^{n}\binom{r+t}{r}=\binom{r+n+1}{r+1}$$
The two series are similar in the sense that the lower number of binomial coefficient in both is a constant. However , I can't get rid of the variable multiplication of $t$ to the binomial coefficient in the first series. How can I proceed further or are there other methods to solve it ?