In the Canadian province Ontario, license plates have the form LLL-DDDD, where L
ID: 3864613 • Letter: I
Question
In the Canadian province Ontario, license plates have the form LLL-DDDD, where L represents a position occupied by an uppercaseletter of the English alphabet, and D represents a position occupied by a decimal digit (i.e, 0 through 9). When I say license plates are distinct, I mean no two of them are identical.
(a) How many distinct plates can be issued, if there are no other restrictions ?
(b) How many distinct plates can be issued, if the letters used need not be distinct but the digits used must be distinct ?
(c) How many distinct plates can be issued, if the letters form a palindrome and the digits form a palindrome ?
Explanation / Answer
(a) How many distinct plates can be issued, if there are no other restrictions ?
Ans. As we have to take 3 uppaercase letter, so we can take this in 26 * 26 * 26 ways. As there is no other restriction on this.
And we have to take 4 digits, so we can take in 10 * 10* 10* 10 ways. As there is no other restriction on this.
So total ways is 26 * 26* 26*10*10*10*10 = 175760000.
(b) How many distinct plates can be issued, if the letters used need not be distinct but the digits used must be distinct ?
Ans. In this there is no restriction on letters. so letter can be taken in 26*26*26 ways
but, there is restriction on digits taken. so digits can be taken in 10*9*8*7 ways
So, total ways is 26*26*26*10*9*8*7 = 88583040
(c) How many distinct plates can be issued, if the letters form a palindrome and the digits form a palindrome ?
Ans. Now, here there is restiction on letter taken. i.e. letters form a palindrome.
And we have to take 3 letters. so this can be done in 26*26*1 = 676 ways.
On digits, there is also same restriction. i.e. digits form a palindrome.
And we have to take 4 digits, so this can be done in 10*10*1*1 = 100 ways
So total ways is 676*100= 67600 ways