Cardinality of Set of Bijections

Theorem

Let $S$ and $T$ be finite sets with the same cardinality:

$\size S = \size T = n$

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

Proof

Follows directly from Cardinality of Set of Injections and Equivalence of Mappings between Finite 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

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 19$: Combinatorial Analysis: Theorem $19.7$