Just to sharpen my intuition in combinatorics, I ask you of ways to think about interesting combinatorical quantities and expressions like the binomial coefficient, for example, for the binomial coefficient I know the following
- There are $\binom{n}{k}$ ways to choose k elements from a set of n elements
- There are $\binom{n}{k}$ strings over $\{0,1\}$ with exactly $k$ ones
- There are $\binom n k$ shortest paths in an rectangular grid from $(0,0)$ to $(k, n-k)$.
Are there more?