Question

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

• A. ${}^nC_m$ if $n \ge m$
• B. ${}^nP_m$ if $n \ge m$
• C. $0$ if $n \ge m$
• D. $n \cdot {}^nC_m$ if $n \ge m$

Hint:

Injections imply one-to-one functions, hence permutations are used.

Answer 1

The Correct Option is B

An injection means one-to-one mapping. Each element in $A$ must go to a distinct element in $B$.
So number of injections from $A$ to $B$ is the number of ways to assign $m$ distinct elements to $n$ available spots (without repetition):
This is permutation: $^nP_m = \frac{n!}{(n - m)!}$