How many $3$ digit numbers can we make using the digits $4,5,7$ and $9$ and where repetition is allowed?

We have four digits.

The numbers we need to make have $3$ digits.

There are $4$ choices for the first digit, $4$ choices for the second digit, and $4$ choices for the third digit since repetition is allowed.

Hence, the total number of $3$ digit numbers that can be made is equal to:

$4\times 4\times 4={ 4 }^{ 3 }$

Hence, option C is correct.