Question

Problem 4. You are the CEO of PlatesRUs license plate manufacturing conglomerate. State regulators tell you that each license plate must contain 2 letters (26 possible letters, A through Z) and 3 numbers (10 possible digits, 0 through 9).

a. State regulators also tell you that the numbers and letters cannot repeat (E.g., AB123 is valid, but AA123 is not). How many different plates are possible? (Note: on a license plate, order matters. AB123 is different from BA123.)

b. Regulators have stated new rules: plates still contain 2 letters and 3 numbers, but now the numbers and letters can be in any of the 5 positions. The numbers and letters still do not repeat on these new plates. (E.g., 1A23B is now valid, but 1A13B is not.) How many plates are possible now?

c. Assume now that plates are still designed as in part b, except that the no-letter-or-- number-repeat clause is not enforced any more. What is the probability that a given plate contains the numbers 777 in succession (i.e. right next to each other, with no letters in between)?

Answer 1

(a) 26*26*10*10*10 = 676000

(b) 5!*26*25*10*9*8 = 56160000

(c) (3*26*26*3!)/(5!*676000) = 0.00015