Chapter 9.1, Problem 23E

Solution

To state: The number of different license plates can be formed if no letters or digits are repeated. The plate begins with two digits, followed by two letters and then three digits.

The resultant license plates are $19656000$ .

Given information:

Every license plate begins with two digits, followed by two letters and then three digits.

Explanation:

Consider the given information,

The license plates begin with two digits, followed by two letters and then three digits and no digit is repeated.

There are 26 letters in alphabets and there are 10 digits from 0 to 9. For forming number plates these 26 letters and 10 digits are to be used.

The first two numbers of the plates must be a digit. Then the number of ways in which these places are filled with no repetition is: $10\cdot 9$

The next two places are to be filled with two letters. The number of ways for it with no repetition is: $26\cdot 25$

The last three places are to be filled with three digits. The number of ways for it without repeating the digits is: $8\cdot 7\cdot 6$

Now the total number of ways will become: $10\cdot 9\cdot 26\cdot 25\cdot 8\cdot 7\cdot 6=19656000$

Therefore, $19656000$ license plates can be formed.