Cardinality of Set of Bijections

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Theorem

Let $S$ and $T$ be sets such that $\size S = \size T = n$.


Then there are $n!$ bijections from $S$ to $T$.


Proof

Follows directly from Cardinality of Set of Injections and Equivalence of Mappings between Sets of Same Cardinality.

$\blacksquare$


Examples

Set of Cardinality $4$

Let $S$ be a set whose cardinality is $4$:

$\card S = 4$

Then there are $24$ bijections from $S$ to itself.


Sources