Fundamental Counting Principle How many fourdigit numbers...

Fundamental Counting Principle How many fourdigit numbers can be formed under each condition?
(a) The leading digit cannot be $0 .$
(b) The leading digit cannot be 0 and the number must be less than 5000 .
(c) The leading digit cannot be 0 and the number must be even.

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Key Concepts

Fundamental Counting Principle

This principle states that if one event can occur in m ways and another independent event can occur in n ways, then the two events together can occur in m × n ways. It is used to determine the total number of outcomes for sequential events, such as selecting digits for different positions in a number.

Counting with Restrictions

In many counting problems, certain positions or choices are restricted by given conditions. Whether it is excluding a digit, limiting a range, or enforcing parity (even or odd), this concept focuses on adjusting the number of available choices per position to meet these criteria.

Digit Place Value Constraints

When forming numbers, each digit position (such as the thousands, hundreds, tens, and ones places) plays a unique role, and restrictions can apply differently to each. For instance, the leading digit may have different constraints compared to others, or the last digit might be required to meet a condition such as being even, which directly influences the counting process.

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