How many $3$-digit numbers can be formed from the digits $1,2,3,4$ and $5$ assuming that.

i) Repetition of the digits is allowed?

ii) Repetition of the digits is not allowed?

A three digit number is to be formed from given 5 digits $1,2,3,4,5$.
$$\begin{gathered} \square \square \square \\ HTO \end{gathered}$$
Now, there are $5$ ways to fill ones place.
Since, repetition is allowed , so tens place can also be filled by $5$ ways.
Similarly,hundreds place can also be filled by $5$ ways.
So, number of ways in which three digit numbers can be formed from the given digits is $5\times 5\times 5=125$

Alternative Method:
There will be as many way ways as there are ways of filling $3$ vacant places in succession by the given five digits. In this case, repetition of digits is allowed. Therefore, the units place can be filed  by any of the given $5$ digits.
Similarly, tens and hundreds digits can be filled in by any given $5$ digits.
Thus, by the multiplication principle, the number of ways in which three digit numbers can be formed from the given digits is $5\times 5\times 5=125$

ii) A three digit number is to be formed from given $5$ digits $1,2,3,4,5$.