How many Licence plates can be made using either two or three uppercase English letters followed by either two or three digits?

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How many license plates can be made consisting of 3 letters followed by 2 digits?
How many license plates can be made using either 3 digits followed by 3 letters or 3 letters followed by 3 digits?
How many Licence plates can be made using either two or three uppercase English letters followed by either two or three digits?
How many license plates can be made using either two or three letters followed by either two or three digits?
How many license plates can be made using either two English letters followed by four digits or two digits followed by four English letters?
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Solution:

Given, license plates consist of 3 letters followed by 2 digits.

Let the numbers on license plates be N

Let the letters on license plates be L

So, the license plate consisting of 3 letters and 2 digits will be LLLNN.

Letters can be anything from A to Z.

There are 26 letter combinations for the first letter. Again second and third letters can be anything from the
26 letters.

So, combination for letters = 26 × 26 × 26

= 17576

Numbers can be anything from 0 to 9.

There are 10 combinations for each place.

So, the combination for numbers = 10 × 10 = 100

Now, the combination for letters and numbers = 17576 × 100 = 1757600.

Therefore, 1757600 license plates can be made.

How many license plates can be made consisting of 3 letters followed by 2 digits?

Summary:

1757600 license plates can be made consisting of 3 letters followed by 2 digits.

How many license plates can be made using either 3 digits followed by 3 letters or 3 letters followed by 3 digits?

The sum rule says that the total number of license plates that can be made using either 3 digits followed by 3 capital letters, or 3 capital letters followed by 3 digits is 263 · 103 + 103 · 263 = 2(263 · 103).

How many Licence plates can be made using either two or three uppercase English letters followed by either two or three digits?

How many license plates can be made using either two or three letters followed by either two or three digits?

There are 26 possible letters.

How many license plates can be made using either two English letters followed by four digits or two digits followed by four English letters?

Since A and B are disjoint, using the sum rule, we get, the number of license plates that can have either four digits followed by two letters or two letters followed by four digits is 104 × 262 + 262 × 104 = 13520000.