In a certain state, automobile license plates start with one...

In a certain state, automobile license plates start with one letter of the alphabet, followed by five digits $(0,1,2, \ldots, 9) .$ Find how many different license plates are possible if (a) the first digit following the letter cannot be 0 (b) the first letter cannot be O or I and the first digit cannot be 0

In a certain state, automobile license plates start with one letter of the alphabet, followed by five digits $(0,1,2, \ldots, 9) .$ Find how many different license plates are possible if
(a) the first digit following the letter cannot be 0
(b) the first letter cannot be O or I and the first digit cannot
be 0

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Key Concepts

Fundamental Counting Principle

This principle states that if there are multiple stages in a process and each stage has a certain number of independent options, the total number of outcomes is the product of the number of choices available at each stage. It is a foundational concept in combinatorics used to count the number of ways a sequence of choices or events can occur.

Counting with Restrictions

Many counting problems involve constraints that limit the choices available for one or more stages. This concept involves adjusting the number of outcomes for each stage based on given restrictions, such as excluding certain elements or values, before applying the multiplication principle.

Permutations with Repetition and Constraints

This concept involves counting the number of ordered arrangements where some elements may be repeated, but restrictions might limit the values that can be chosen in specific positions. By considering the number of choices available in each slot, particularly when certain options are not permitted, one can compute the total number of valid arrangements.

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