Since it has to be greater than 3000, and we have 4 digits, we have to use all of the 4 digits.
Lets starts off by with 4 blank boxes. The numbers we put in the boxes will be the number of options in that position.
----- ----- ----- -----
| | | | | | | |
----- ----- ----- -----
Firstly we will look at our two restrictions.
Now for it to be greater than 3000, the first digit has to be either 4,5 or 7, so we have three options.
----- ----- ----- -----
| 3 | | | | | | |
----- ----- ----- -----
Now for the number to be odd, the last digit has to be an odd number, this is where the situation splits. If the first digit was odd then the first digit only had two options we will only have 1 option left, if the first digit was even (meaning it had 1 option, as it has to be the 4) then we have 2 options left, the 5 or 7.
So we have two situations.
----- ----- ----- ----- ----- ----- ----- -----
| 2 | | | | | | 1 | | 1 | | | | | | 2 |
----- ----- ----- ----- ----- ----- ----- -----
Now for the two middle numbers, for the first option we have two digits left, and once we use one of those we will only have 1 left. So we have a 2 and a 1.
----- ----- ----- ----- ----- ----- ----- -----
| 2 | | 2 | | 1 | | 1 | | 1 | | 2 | | 1 | | 2 |
----- ----- ----- ----- ----- ----- ----- -----
So the number of options is
+\left(1\times2\times1\times2\right)=4+4=8)
Spoiler
And to check the answer I went through the options, not the best idea but to be sure

4275
4725
4257
4527
5247
5427
7245
7425
Hope that helps, there's probably a simpler way of doing it though.