I read an article about Viterbi algorithm on wikipedia today, which (somehow) reminded me of interesting combinatorics problems back in the day. So I will share some of my favorite ones.

Give an intuitively understandable description to the following combinatorics expressions:

$$1. _n C _k = _{n-1}C _{k} + _{n-1} C _{k-1} \\ 2. k_n C _k = n _{n-1} C _{k-1} \\ 3. \sum _{r=k}^{n}{_{r} C _{k}} = _{n+1} C _{k+1}$$