Question

In a certain state, license plates cach consist of 2 letters followed by 3 digits. (a) How many different license plates are there? There are only upper case letters. (b) How many different license plates are there that have no repeated letters or digits?

Answer 1

Answer: (a) 676000

(b) 468000

Explanation:

We know that the total number of digits in the number system from 0 to 9= 10

The total number of letters in English alphabet (Only upper case) from A to Z = 26

Given : In a certain state, license plates cach consist of 2 letters followed by 3 digits.

(a) If repetition is allowed , then the total number of different license plates there are :-

`26*26*10*10*10=676000`

hence, the number of different license plates = 676000

(b) If repetition is not allowed , then the total number of different license plates there are :-

`^(26)P_2*^(10)P_3\\\\=(26!)/((26-2)!)*(10!)/((10-3)!)\\\\=(26*25*24!)/(24!)*(10!)/(7!)\\\\=468000`

Hence, the number of different license plates are there that have no repeated letters or digits=468000