Solution for Problem 8.6.4

License plates for cars have to be unique. If a license plate contains 6 characters consisting of 2 letters followed by 4 digits  example:  QW2354

  • how many different license are possible?

  • how many different license are possible if letters were allowed to repeated but numbers are not allowed to be repeated?

Solve Step
How many different license are possible?

There are 26 × 26 × 10 × 10 × 10 × 10
 =  6,760,000 different plates.  

Multiplication Principle
26 letters, 10 digits 0 - 9
Letters repeated but not numbers...

 26 × 26

Start with 26 letters . Repeats allowed such as AA or BB etc...

26 × 26 × 10 × 9 × 8 × 7 = 

Repeating numbers are not allowed.
1174, 1231, 1111 are not allowed.

If the first 3 characters are  AA4, then the next number can't be 4.  There are only 9 numbers left to choose from.  If the next number is 2, AA42, there are only 8 numbers left to choose from and so on....

There are 26 × 26 × 10 × 9 × 8 × 7 =  3,407,040 different plates with no repeating numbers.