(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.