Exercise 444Combinatorics › Counting principlesHigh schoolAdded 3 Nov 2026

Counting injections

#injections#arrangements#permutations

Statement

Count injective functions from an $m$ -element set to an $n$ -element set (with $n\ge m$ ).

Hint

The first image has $n$ choices, then $n-1$ , etc.

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Product. $n(n-1)\cdots(n-m+1)=\frac{n!}{(n-m)!}=A_n^m$ .

Conclusion. There are $\frac{n!}{(n-m)!}$ injections.

Example. $m=3$ , $n=5$ : $5\cdot4\cdot3=60$ injections.