Let:

Domain size = 3 elements
Codomain size = 5 elements

To create a one-to-one (injective) function, every element in the domain must map to a distinct element in the codomain.
The number of such functions is given by the number of injective mappings: \[ ^5P_3 = \frac{5!}{(5 - 3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 60 \]