Combinatorial sum within interval.

Question

How can I compute :

$\sum_{j = \alpha}^{\beta} \binom{\beta}{j}$, for $\alpha < \beta$ and $\alpha, \beta \in \mathbb{N}$ ?

Answer 1

We have that

$$\sum_{j = \alpha}^{\beta} \binom{\beta}{j}=\sum_{j = 0}^{\beta} \binom{\beta}{j}-\sum_{j = 0}^{\alpha-1} \binom{\beta}{j}$$

but for the partial sum we don't have a closed formula. Refer for example here.