Final answer:
There are 604,800 unique 7-digit ZIP codes that can be created with no repeat digits by calculating the permutations of the first seven counting numbers starting from 10.
Step-by-step explanation:
Calculating the Number of Unique 7-Digit ZIP Codes Without Repeats
To calculate the number of 7-digit ZIP codes possible without any repeats, you would use permutations because the order of the digits matters and each digit can be used only once. There are 10 possible digits (0-9) for each place in a ZIP code. For the first digit, there are 10 options, for the second digit there are 9 remaining options (as one has been used for the first digit), and this pattern continues decreasing by one each time. Therefore, the calculation for the number of possible ZIP codes is 10 × 9 × 8 × 7 × 6 × 5 × 4, which is the product of the first seven counting numbers starting from 10 and decreasing by one each time.
The number of different 7-digit ZIP codes without repeats is 604,800.