.
If repetition is allowed then how many different three digits numbers can be formed
using the digits from 1 to 5?
~~~~~~~~~~~~~~~~~~~
(1) First, let assume that all three digits in a three-digit numbers are different,
and ask ourselves how many such three-digit numbers are possible.
The answer is clear and elementary: there are 5*4*3 = 60 such three-digit numbers.
(2) Next, let assume that exactly two digits in our three-digit number are identical.
How many such three-digit numbers are possible ?
In this situation, we can select the unique digit in 5 different way, and then
we can select the repeating digit in 4 different ways, which gives 5*4 = 20 different selections.
With these digits (one unique and two others are repeating), we can make/produce 3 permutations,
which give us 20*3 = 60 different three-digit numbers of this kind.
(3) Finally, having five digits 1, 2, 3, 4, 5, we can form 5 (five) three-digit numbers,
where all three digits are identical.
(4) Summing up all these options, we find that the answer to the problem's question
is 60 + 60 + 5 = 125 different three-digit numbers, satisfying the imposed conditions.
ANSWER. In all, there are 125 different three-digit numbers, satisfying the imposed conditions.
Solved.
---------------
Alan got this result in much simpler way (!)