If we fix first 2 digits we'll get exactly 7*8 numbers with this proprty. So the number of non-repeaters between 1500 and 8000 is equal to the number of non-repeaters between 15 and 80 multiplied by 56. So:

56(5+9+9+9+9+9+9)=56.59=3304.

Another way:

First we'll calculate the number of non-repeaters between 1000 and 1500:

We have 6 possibilities for the second digit but it's unequal to 1 and 5 so there are 4 real possibilities. For the other two digits we have 8.7 possibilities. So total 8.7.4=224 numbers between 1000 and 1500

Analogous we have 7.9.8.7=3528 non-repeaters between 1000 and 8000 so we have 3528-224=3304 non-repeaters between 1500 and 8000.