Strictly increasing maps

Question

For $p\ge n$, how many strictly increasing maps from $N^*_n$ to $N^*_p$ do exist, where $N^*_n = \{1, 2, \dots, n\}$ is the set of the first $n$ integers greater than 0 ?

My answer: uncountable many. Is this correct?

Thanks for any help!

Answer 1

Answer. $\binom{p}{n}$.

Every such map is fully characterized by its range, and its range is a subset of $N_p^*$ consisting of $n$ elements. There are $\binom{p}{n}$ such subsets.