I've come across many problems that require me to find summation of binomial coefficients.
How go about solving these kind of summations of the form $$\sum_{r=0}^{n} \binom{n}{r}f(r)$$ where f(r) is some function
for starters how to do these
$$\sum_{r=0}^{n} \binom{n}{r}(r+1)$$
$$\sum_{r=0}^{n} \binom{n}{r}\frac{r^2}{3}$$
I remember someone using differentiation somewhere but I can't find where it was. So I thought this would be a good place where peiple can look