# Ways to think about the binomial coefficient

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?

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There are many, many "combinatorial quantities and expressions like the binomial coefficient," and even more ways to think about them all. – anon Jul 17 '12 at 21:05
oeis.org/A007318 – datageist Jul 17 '12 at 21:10