Question

License plates in a particular state are to consist of 2 digits followed by 3 uppercase letters. ​ ​c) How many different license plates can there be in this state if the must be ​, and repetition of letters and numbers is not​ permitted? ​d) How many different license plates can there be in this state if the first digit cannot be ​, and repetition of letters and numbers is not​ permitted?

Answer 1

c) 1560000 possible license plates if the first digit can be 0 and repetition is not permitted.

d) 1404000 possible license plates if the first digit cannot be 0 and repetition is not permitted.

License Plate Combinations

Let's analyze the number of possible license plates for each scenario:

c) No restrictions on first digit:

Digits: 10 choices (0-9) for each of the 2 positions, resulting in 10 * 10 = 100 possibilities.

Letters: 26 choices (A-Z) for the first letter, 25 choices for the second letter (excluding the one used previously), and 24 choices for the third letter (excluding the two used previously), resulting in 26 * 25 * 24 = 15600 possibilities.

Total: 100 * 15600 = 1560000 possible license plates.

d) First digit cannot be 0:

Digits: 9 choices (1-9) for each of the 2 positions, resulting in 9 * 9 = 81 possibilities.

Letters: Same as scenario c, with 26 choices for the first letter, 25 choices for the second, and 24 choices for the third, resulting in 15600 possibilities.

Total: 81 * 15600 = 1404000 possible license plates.

Therefore, there are:

c) 1560000 possible license plates if the first digit can be 0 and repetition is not permitted.

d) 1404000 possible license plates if the first digit cannot be 0 and repetition is not permitted.