vc dimension – Understanding growth function of closed intervals in $mathbb{R}$


I as studying VCdimensions and growth functions and found the following example on Wikipedia:

The domain is the real like $mathbb{R}$. The set H contains all the real intervals, i.e., all sets of form ${c in (x_1, x_2) | x in mathbb{R}}$ for some $x_{0, 1} in mathbb{R}$.

For any set C of m real numbers, the intersection $H cap C$ contains all runs of between 0 and m consecutive elements of C. The number of such runs of ${m+1 choose 2} + 1$, so Growth(H, m) = ${m+1 choose 2} + 1$.

Can anyone please explain to me what does the term “all runs of between 0 and m” refer to here and why the growth function is ${m+1 choose 2} + 1$ and not ${m+1 choose 2}$?

Thank you very much!