Corollary 2

Let E and F be totally ordered finite sets with $p$ and $n$ elements, respectively. Then the number of strictly increasing mappings of E into F is $\binom{n}{p}$.

For such a mapping is an injection of E into F (§ 1, no. 12, Proposition 11), and since E and F are well-ordered (§ 4, no. 4, Corollary 1 to Proposition 3), for each subset X of $p$ elements of F there is exactly one strictly increasing mapping of E onto X (§ 2, no. 5, Theorem 3).