Question

License plate numbers in a certain state consists of seven characters. The first character is a non-zero digit (1 through 9). The next four characters are capital letters (A through Z) and the last two characters are non-zero digits. Therefore, a license plate number in this state can be any string of the form:

Digit-Letter-Letter-Letter-Letter-Digit-Digit

(a) How many different liscence plate numbers are possible?
(b) How man licence plate numbers are possible if no digit appears more than once?
(c) How man licence plate numbers are possible if no digit or letter appears more than once?

Answer 1

a) 333,135,504 different plates

b) 230,315,904 different plates

c) 180,835,200 different plates

Explanation:

Pattern: Digit(1-9)-Letter-Letter-Letter-Letter-Digit(1-9)-Digit (1-9)

We will calculate the number of possibilities for the digits part, then for the letters part, then we'll multiply them together.

For the digits, we have 3 numbers, first and last 2 positions. We can consider this is a single 3-digit number, where n = 9 (since they are non-zero digits) and r = 3.

For the letters part, it's basically a 4-letter word, where n = 26 (A through Z) and r = 4.

(a) How many different license plate numbers are possible?

No limitation on repeats for this question:

For the digits, we have 9 * 9 * 9 = 729 (since repetition is allowed, and we can pick any digit from 0 to 9 for each position)

For the letters we have: 26 * 26 * 26 * 26 = 456,976

Because the digits and letters arrangements are independent from each other, we multiply the two numbers of possibilities to have the global number of possibilities:

P = 729 * 456976 = 333,135,504 different plates, when there's no repeat limitation.

(b) How man license plate numbers are possible if no digit appears more than once?

Repeats limitation on digits:

For the digits, we have 9 * 8 * 7 = 504 (since repetition is NOT allowed, we can pick any of 9 digits for first position, then any 8 remaining and finally any 7 remaining at the end)

For the letters we still have: 26 * 26 * 26 * 26 = 456,976

Because the digits and letters arrangements are independent from each other, we multiply the two numbers of possibilities to have the global number of possibilities:

P = 504 * 456976 = 230,315,904 different plates, when there's no repeat on the digits.

(c) How man license plate numbers are possible if no digit or letter appears more than once?

Repeats limitation on both digits and letters:

For the digits, we have 9 * 8 * 7 = 504 (

For the letters we still have: 26 * 25 * 24 * 23 = 358,800

Because the digits and letters arrangements are independent from each other, we multiply the two numbers of possibilities to have the global number of possibilities:

P = 504 * 358800 = 180,835,200 different plates, when there's no repeat on the digits AND on the letters.