Q:

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?

Accepted Solution

A:
Answer: (a) 676000(b) 468000 Step-by-step explanation:We know that the total number of digits in the number system from 0 to 9= 10The total number of letters in English alphabet (Only upper case) from A to Z = 26Given : 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 :-[tex]26\times26\times10\times10\times10=676000[/tex]hence, the number of different license plates = 676000(b)  If repetition is not allowed , then the total number of different license plates there are :-[tex]^{26}P_2\times^{10}P_3\\\\=\dfrac{26!}{(26-2)!}\times\dfrac{10!}{(10-3)!}\\\\=\dfrac{26\times25\times24!}{24!}\times\dfrac{10!}{7!}\\\\=468000[/tex]Hence, the number of  different license plates are there that have no repeated letters or digits=468000