Think about it this way. Take each of the students into a separate room and ask her or him: which tire went flat? Each student has four choices. If they haven't planned an answer to the question, each must choose one of four possibilities. How many different ways can the four make their replies?
Let's call the tires ABCD. Then, in no particular order because order isn't important, the students might have answered DDAB or ABDC or even AAAA. How many such possibilities are there?
Here's a similar question that will help us understand: how many 4 digit numbers are there (any digit may be 0-9)? That should be easy: you can go from 0000 to 9999, so there are 10 000 such numbers. In general, there are 10 choices for each digit and in this case, four digits in all, so that is 104.
In the case of the tires, there are four choices for each student and four students in all. So now you can probably figure out how many possibliities there are.
Now that you know how many possible answers the group of students could give, ask yourself how many of those actually all indicate the same tire (there's more than one!). The ratio of this last value divided by the total possibilities is, of course, the probability you are trying to figure out.
Hope this helps!