Question

How many license plates can be made with 3 letters followed by 3 digits exist if exactly one of the digits is 1? Repetition of either letter or digit is permitted.

Example of a valid plate: AAA122

Answer 1

The first 3 places are Letters . The letters can be repeated. So for the 1st place there are 26 letters that can be placed. So 26 ways. Similarly for 2nd and 3rd place there is also 26 ways each. So only for letters number of ways is 26×26×26 .

Now for the last 3 places out of 3 places one is 1 and the remaining 2 can be any numbers from 0,2,3,••••,9 (not 1) . That is there is 9 numbers from which the remaining 2 places can be filled. So number of ways to fill the 2 places of number section is = 9×9

Now 1 can be placed at any place of the last 3 places. i.e fourth fifth or sixth place. So number of ways to place 1 is = 3

So total number of different license plate is

=26×26×26×9×9×3 = 4270968