Problem Solving

Source: Manhattan Prep

A license plate in the country Kerrania consists of four digits followed by two letters. The letters A, B, and C are used only by government vehicles while the letters D through Z are used by non-government vehicles. Kerrania's intelligence agency has recently captured a message from the country Gonzalia indicating that an electronic transmitter has been installed in a Kerrania government vehicle with a license plate starting with 79. If it takes the police 10 minutes to inspect each vehicle, what is the probability that the police will find the transmitter within three hours?

A. 18/79
B. 1/6
C. 1/25
D. 1/50
E. 1/900

The OA is D

If it takes 10 minutes to inspect one vehicle, the number of vehicles that can be inspected in 3 hours (180 minutes) = 180/10 = 18. Hence, for calculating the probability that the police will find the transmitter within three hours, the favorable cases = 18. Now, we need to figure out the total # of cases.

The total number of cases = Total cars possible. The details given about the car is that it starts with 79, which leaves 2 more digits, both of which can be filled by all 10 numbers (0-9).
Given that vehicle to be found is government's , we have 3 letters, each of which can be filled by any from the set {A,B,C}. Hence the total number of such cars possible = 10*10*3*3 = 900

So, the probability that the police will find the transmitter in 3 hours
18/900 = 1/50. Option D

The license plate consists of 4 digits followed by 2 letters
The letters A, B, and C are used by government vehicles
The letters D and Z are used by non-government vehicles


The intelligence agency is looking for a license plate starting with 79. Hence, there are 2 unknown digits and 2 unknown letters.
It takes the police 10 minutes to search 1 car so in (3 * 60 = 180) minutes, they can search 180/10 = 18 cars


For 2 unknown digits, there are 10 possible digits from 0 to 9
For 2 unknown letters, there are 3 possible letters A, B, and C for government vehicles
Therefore, total possible plates = 10 * 10 * 3 * 3 = 900
The probability of finding the transmitter in 3 hours = 18/900
= 1/50


Answer = D