Counting Functions
How many total functions are there from set \(A\) to set \(B\) if \(|A|=3\) and \(|B|=7\)? Please enter your answer in the form x^y.
Say \(A = \{a_1, a_2, a_3 \}\). There is a bijection between total functions from \(A\) to \(B\) and length-3 vectors of elements of \(B\).
In this bijection, a given total function \(f\) corresponds to the vector \((f(a_1), f(a_2), f(a_3)) \in B^3\).
Hence, by the product rule, \(|B^3| = 7^3\).