Plate Numbers How many different license plate numbers can...

Plate Numbers How many different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9 , if:
(a) Letters and digits may be repeated?
(b) Letters may be repeated, but digits may not?
(c) Neither letters nor digits may be repeated?

Key Concepts

Fundamental Counting Principle

This principle states that if there are several independent choices to be made, the total number of outcomes is the product of the number of ways each individual choice can be made. It is the foundation for solving counting problems by multiplying the number of choices available at each step.

Permutations with Repetition

Permutations with repetition occur when choices for each position can be repeated or reused. In these scenarios, every position has the same number of possibilities regardless of the previous choices, making the total number the product of the number of choices raised to the power of the number of positions.

Permutations without Repetition

Permutations without repetition are used when each choice can only be selected once, which means the number of available options decreases with each position filled. This leads to a product of decreasing factors, often represented with factorials or partial factorials, and is essential when arranging distinct items in order.

Counting Arrangements with Mixed Restrictions

Many problems involve arrangements where different segments follow different repetition restrictions. The key is to separately apply the appropriate counting method for each segment—for example, one may allow repetition for one segment and disallow it for another—and then combine the results using the Fundamental Counting Principle.

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