There are 1,200 sequences out of 3,125 that satisfy your requirement of two similar numbers and three distinct ones being drawn. To see this, first figure the number of ways that persons 1 and 2 can draw the same number and persons 3, 4 and 5 each draw a number that is different from those already drawn. Then think about in how many ways each of those sequences can be arranged.

Your description of what you have tried suggests that you got the right answer but for the wrong reason.