# Cardinality of $[\lambda]^\kappa$

Let $$\kappa \leq \lambda$$ cardinals with $$\lambda$$ infinite, and $$[\lambda]^\kappa=\{Y\subseteq\lambda : ot(Y,\in)=\kappa\}$$. I want to show that $$[\lambda]^\kappa \asymp\ ^\kappa\lambda$$.

I've aldredy proved that $$[\lambda]^\kappa \asymp \{f \in \ ^\kappa\lambda: f\ \text{is increasing}\}$$, and so easily $$[\lambda]^\kappa\preceq \ ^\kappa\lambda$$.

For the other injective function I don't have any ideas: tryng to associate the function to its range dosen't work, because I find only subset with cardinality $$\kappa$$ but not with order type $$\kappa$$.

Do you have some suggestions?