How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$?
This counts the number of subsets from $\{1,2,3,\dots,n\}$ having size $7$. To me, the summation part counts subsets of size $6$. Can someone please tell me how this is done?
P.S. I don't know how to edit in the proper way.
Suppose you want to pick $7$ elements from $\{1, 2, 3, \dots, n\}$. One way to do this is to first pick the largest of the $7$ elements (call it $k$), and then pick the smaller $6$ elements from $\{1, 2, 3, \dots, k-1\}$. Counting this separately for all choices $k$ gives
$$ \binom{n}{7} = \sum_{k=7}^{n} \binom{k-1}{6}$$
as required.
(About the lower bound of $k=7$: You can see that $k$ must be at least $7$ as it's the largest of $7$ elements, or you can see that $\binom{k-1}{6}$ is nonzero only when $k \ge 7$.)
