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(B) An injection (or one-to-one function) from set $A$ to set $B$ exists only if the number of elements in $A$ is less than or equal to the number of

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$^nP_m$ if $n \ge m$

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(B) An injection (or one-to-one function) from set $A$ to set $B$ exists only if the number of elements in $A$ is less than or equal to the number of elements in $B$,i.e.,$m \le n$.If $n If $n \ge m$,we need to choose $m$ distinct elements from $B$ and arrange them in a specific order to map to the $m$ elements of $A$.The number of ways to do this is given by the permutation formula $^nP_m = \frac{n!}{(n-m)!}$.Thus,the number of injections is $\begin{cases} ^nP_m, & n \ge m \ 0, & n < m \end{cases}$.

If a set $A$ has $m$ elements and the set $B$ has $n$ elements,then the number of injections from $A$ to $B$ is

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