Problem 87 (difficulty: 2/10)

Let \(\displaystyle A=\{ 1,2,...,n \}\) and \(\displaystyle B=\{ 1,...,k \}\).

 (1)  How many different functions \(\displaystyle f:A\to B\) do exist?
 (2)  How many different injective functions \(\displaystyle f:A\to B\) do exist?
 (3)  How many different functions \(\displaystyle f:A_{0}\to B\) do exist, where \(\displaystyle A_0\subset A\) is arbitrary?

Answer:

 (1)  \(\displaystyle |B|^{|A|}=k^n\).
 (2)  \(\displaystyle \binom{k}{n}\cdot n!=k(k-1)\cdots(k-n+1)\).
 (3)  \(\displaystyle (|B|+1)^{|A|}=(k+1)^n\).

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