EXAMPLE 1.4.7

In Florida, standard automobile license plate "numbers" used to follow the following scheme: 3 letters -- 2 digits -- 1 letter

Examples: JKP47R TRR39S VWN22Y ZQW05Z

1. How many different license plate codes were possible under this scheme?

SOLUTION

In order to create a license plate code we need to make 6 decisions:

i. Choose first letter: 26 options;

ii. Choose second letter: 26 options;

iii. Choose third letter: 26 options;

iv. Choose first digit: 10 options;

v. Choose second digit: 10 options;

vi. Choose last letter: 26 options.

According to the Fundamental Counting Principle the number of outcomes is:

(26)(26)(26)(10)(10)(26) = 45,697,600

This might seem like a ridiculously large number at first, but to put it in perspective take into account the fact the there are more than 12 million people in Florida and there are more cars than people. In that case, 46 million different license plate codes is not an extremely large number.

2. Now they follow this scheme:

2 letters -- 1 letter or digit -- 2 digits -- 1 letter

Now how many codes are possible?

SOLUTION

Again, in order to create a license plate code we need to make 6 decisions:

i. Choose first letter: 26 options;

ii. Choose second letter: 26 options;

iii. Choose third symbol (letter or digit): 36 options;

iv. Choose first digit: 10 options;

v. Choose second digit: 10 options;

vi. Choose last letter: 26 options.

According to the Fundamental Counting Principle the number of outcomes is:

(26)(26)(36)(10)(10)(26) = 63,273,600

3. Why take such a piecemeal approach to expanding the number of possible codes?

You might wonder, if there aren't enough license codes, why not employ a new scheme that provides a huge number of new codes? Rather than expanding from 46 million to 63 million codes, why not use a method that would expand from 46 million to 500 million codes, or from 46 million to 500 billion codes?

One answer is that the more possible codes we have, the more difficult it is to keep track of them all. For example, we know that license plate codes are "censored" in the sense that certain letter-number combinations are screened out if they might be deemed offensive or controversial. The greater the number of possible codes, the more difficult this screening process becomes.