Cardinality of collection of all subsets whose cardinality is smaller than the original set

Solution 1:

What you're trying to do is flatly false, unless $|X|=\aleph_0$.

Suppose that $\aleph_0<|X|$, then $[X]^{<|X|}$ contains all the countable subsets of $X$, so its cardinality is at least $|X|^{\aleph_0}$, but that is at least as large as $2^{\aleph_0}$, and indeed equal to it under $\sf MA$.

In general we can't say lot about $[\kappa]^{<\kappa}$, only that its size is exactly $\kappa^{<\kappa}$, which can be very different between models of $\sf ZFC$. One thing though, is that if $\kappa=\lambda^+$, then $\kappa^{<\kappa}=\kappa^\lambda=2^\lambda$. Another is that if $\kappa$ is singular, then $\kappa<\kappa^{<\kappa}$.