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So far we have not used a function as a model for binomial coefficients (combinations). Think for a moment about the relationship between combinations and permutations, say specifically \({9 \choose 3}\) and \(P(9,3)\text{.}\) We do have a function model for \(P(9,3)\text{.}\) This is the number of injective functions from a set of size 3 (say \(\{1,2,3\}\) to a set of size 9 (say \(\{1,2,\ldots, 9\}\)) since there are 9 choices for where to send the first element of the domain, then only 8 choices for the second, and 7 choices for the third. For example, the function might look like this:

\begin{equation*} f(1) = 5 \qquad f(2) = 8 \qquad f(3) = 4\text{.} \end{equation*}
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