Question

A state's license plate starts with a digit other than 0 followed by four capital letters (A through Z) followed by two digits ( through 9). (a) How many different license plates are possible? (b) How many different license plates start with a 2 and end with a 9?(c) How many different license plates have no repeated symbols (all the digits are different and all the letters are different)?

Answer 1

(a) There are 9 possibilities for the first digit (1-9), 26 possibilities for each of the four capital letters (A-Z), and 10 possibilities for each of the last two digits (0-9). So, the total number of different license plates is:

9 * 26 * 26 * 26 * 26 * 10 * 10 = 9 * (26^4) * 100 = 152,409,600

(b) If the plate starts with a 2 and ends with a 9, there are still 26 possibilities for each of the four capital letters. So, the total number of different license plates with these conditions is:

1 * 26 * 26 * 26 * 26 * 1 * 1 = (26^4) = 456,976

(c) For a license plate with no repeated symbols, there are 9 possibilities for the first digit (1-9), 26 possibilities for the first letter, 25 for the second letter, 24 for the third letter, and 23 for the fourth letter. For the last two digits, there are 10 possibilities for the first digit, and 9 for the last digit. So, the total number of different license plates with no repeated symbols is:

9 * 26 * 25 * 24 * 23 * 10 * 9 = 135,274,080

Answer 2

There are 45,697,600 different possible license plates. Only 456,976 license plates can start with a 2 and end with a 9. There are 27,843,750 license plates with no repeated symbols.

Step-by-step explanation:

(a) To calculate the number of different license plates that are possible, we need to consider the number of options for each position on the license plate. The first digit can be any digit other than 0, so there are 9 possibilities. The next four positions can each be any capital letter (A-Z), so there are 26 possibilities for each position. The last two positions can be any digit (0-9), so there are 10 possibilities for each position.

To calculate the total number of different license plates, we multiply the number of possibilities for each position:

Total number of different license plates = 9 * 26 * 26 * 26 * 26 * 10 * 10 = 45,697,600

(b) If we want to find the number of different license plates that start with a 2 and end with a 9, we can fix the values for the first and last positions as 2 and 9, respectively. The remaining positions can still have any digit or letter, so the number of possibilities for each position remains the same. We calculate the total number of different license plates:

Total number of different license plates = 1 * 26 * 26 * 26 * 26 * 10 * 1 = 456,976

(c) To find the number of different license plates with no repeated symbols, we need to consider the choices for each position. The first digit can be any digit other than 0, so there are 9 possibilities. The next four positions can each be any of the remaining 25 letters (we exclude one letter for each position as it cannot be repeated), so there are 25 possibilities for each position. The last two positions can be any of the remaining 9 digits (we exclude one digit for each position as it cannot be repeated), so there are 9 possibilities for each position. We calculate the total number of different license plates:

Total number of different license plates = 9 * 25 * 25 * 25 * 25 * 9 * 9 = 27,843,750