1.1.9 problem 11

(a)

Each of the $n$ inputs has $m$ choices for an output, resulting in $$m^n$$

possible functions.

(b)

If $n<m$, at least two inputs will be mapped to the same output, so no one-to-one function is possible.

If $n\ge m$, the first input has $m$ choices, the second input has $m-1$ choices, and so on. The total number of one-to-one functions then is

$$m(m-1)(m-2)\dots (m-n+1)$$