Question

A typical California license plate consists of a 7-character string as in the image below. The string startswith a number, is then followed by three letters from the alphabet, and then followed by three more numbers.Count the number of possible different license plates that can be formed in this fashion. How many different license plates have the same exact letter permutations (e.g. ‘7XHL245’ and ‘3XHL901’ have the same letters,? How many different license plates are there that contain a vowel for the firstletter and have an odd number in the last slot and the first slot does not contain a 6?

Answer 1

For this question we are going to use Fundamental Principle of Counting : It states that if there are m ways to do an action and n ways to do another action than there are m*n ways of doing both the actions together .

There are 7 places we have to fill up for a license plate so

1st place has 10 different single digit to fill up i.e 0,1,2,3,4,5,6,7,8,9

2nd place has 26 alphabets to fill up

similarly amount of options to fill up each place is :

number in each place refers to no of options we have to fill that place

sdc

So using Fundamental principle of counting we can say that total no of possible license plates are = 10*26*26*26*10*10*10

= 175760000

(b) Now we need no of license plates that have same alphabets but different numbers

  

Since alphabets in the middle are fixed in some order and same alphabets we can only change numbers

   therefore, Total license plate of this type = 10*10*10*10 = 10000

(c) Now the conditions are vowel for first letter , odd numbers in last slot , first slot does not have 6

   So we only have 5 choices for first letter , only 5 choices for last slot that are odd , and 9 choices for first slot since we can't use 6

So total license plate is = 9*5*26*26*10*10*5 = 15210000

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