Which version of the Pigeonhole principle is correct? One is far stronger than the other One version of the Pigeonhole principle says that if the cardinality of a set $A$ is greater than that of a set $B$, then there can be no one-to-one function that maps from $A$ to $B$. 
Another version says: If $n$ elements are partitioned into $m$ subsets, then at least one subset must contain at least $\lceil n/m \rceil$ elements. Note that the $\lceil \ \rceil$ enclosing the $n/m$ tells us to round the value up.
The latter version is just a more powerful version, no? It seems strange to me that one version of a theorem can objectively give more information than another, so it's likely that either the latter version is not correct or I am incorrect in thinking that it somehow has more utility.
Or is this normal? Is it common for certain versions of theorems to simply have less utility than other versions?

Besides the two good answers already present, I would like to mention that the second version is also a trivial corollary of the first: indeed, if you take $m$ disjoint subsets $A_1,\ldots,A_m$ of $A$,  each with less than $\lceil n/m\rceil$ elements, then the union $\bigcup_{j=1}^m A_j$ has at most  $m\lfloor n/m\rfloor<n$ elements; so by the "simple" Pigeonhole Principle there cannot be an injection $A\to\bigcup_{j=1}^m A_j$.
So, for finite sets, both versions are easily seen to be equivalent. As mentioned, the first one also applies to infinite sets. 
