Suppose that the format for license plates in a certain state is two letters followed by four numbers. (a) How many different plates can be made? (b) How many different plates are there if the letters can be repeated but no two numbers can be the same? (c) How many different plates can be made if repeti- tions of numbers and letters are allowed except that no plate can have four zeros?

(a) 6,760,000 plates

(b) 3,407,040 plates

(c) 6,084,000 plates

Explanation:

The very first thing to note about this question is the number of characters involved in the license plate format (6 characters in this case; 2 letters and 4 numbers). The letters come first and then the numbers follow.

There are a total of 26 possible letters (A-Z) and 10 possible numbers (0 - 9) that can be chosen. We can then proceed to the first question;

(a) Here, the total number of possible plates is to be determined. This is done as follows:

Character 1 (Letter): There are 26 possible letters

Character 2 (Letter): There are 26 possible letters

Character 3 (number): There are 10 possible numbers

Character 4 (number): There are 10 possible numbers

Character 5 (number): There are 10 possible numbers

Character 6 (number): There are 10 possible numbers

So, total number of different plates will be obtained by multiplying all the possibilities: 26 × 26 × 10 × 10 × 10 × 10 = 6,760,000 plates

(b) This second part puts a constraint on the usage of the numbers, unlike the question (a), where there was no constraint at all.

Since there is no constraint on the letters, we can write that:

Character 1 (Letter): There are 26 possible letters

Character 2 (Letter): There are 26 possible letters

For the first number as well, we can write:

Character 3 (number): There are 10 possible numbers

However, for the remaining characters, the possibilities will continually reduce by a value of 1, since we can not use a number that has been used before. So,

Character 4 (number): There are 9 possible numbers

Character 5 (number): There are 8 possible numbers

Character 6 (number): There are 7 possible numbers

So, total number of different plates will be: 26 × 26 × 10 × 9 × 8 × 7 = 3,407,040 plates

(c) Here, repetitions are allowed as in questions (a), but there can not be four zeros. This implies that the maximum number of zeros in any plate will be three. Thus, there will be maximum possibilities on all characters until the last one which will be constrained.

Character 1 (Letter): There are 26 possible letters

Character 2 (Letter): There are 26 possible letters

Character 3 (number): There are 10 possible numbers

Character 4 (number): There are 10 possible numbers

Character 5 (number): There are 10 possible numbers

Character 6 (number): There are 9 possible numbers

Total number of plates will therefore be: 26 × 26 × 10 × 10 × 10 × 9 = 6,084,000 plates.