For each of the four subsets of the two properties (a) and...

For each of the four subsets of the two properties (a) and (b), count the number of four-digit numbers whose digits are either $1,2,3,4$, or $5:$ (a) The digits are distinct. (b) The number is even. Note that there are four problems here: $\emptyset$ (no further restriction), \{a\} (property (a) holds), \{b\} (property (b) holds), $\{a, b\}$ (both properties (a) and (b) hold).

For each of the four subsets of the two properties (a) and (b), count the number of four-digit numbers whose digits are either $1,2,3,4$, or $5:$
(a) The digits are distinct.
(b) The number is even.
Note that there are four problems here: $\emptyset$ (no further restriction), \{a\} (property
(a) holds), \{b\} (property
(b) holds), $\{a, b\}$ (both properties (a) and (b) hold).

Key Concepts

Properties of Even Numbers

In number theory, an even number is defined as a number divisible by 2, which fundamentally depends on the digit in the unit's place. In counting problems, applying this property restricts the possible choices for the last digit, thereby affecting the count for sequences or numbers that are required to be even.

Constraints in Combinatorics

This concept involves adjusting counting methods to account for specific restrictions, such as requiring items to be distinct or meet certain criteria (for example, a digit must be even). Handling such constraints often means reducing the available choices at certain steps or dividing the problem into different cases.

Case Analysis in Counting

This technique involves breaking a problem into separate, distinct cases based on the conditions imposed (such as applying one or more restrictions). By solving each case independently and then combining the results, one can manage complex counting scenarios systematically.

Fundamental Counting Principle

This principle states that if there are several steps in a process and each step can be performed independently in a certain number of ways, then the total number of outcomes is the product of the number of ways each individual step can occur. It is the basis for many counting problems where each decision made at a step multiplies with the next.

Permutations and Arrangements

Permutations refer to the arrangement of a set of items where the order matters. When the items (or digits) need to be arranged to form numbers or sequences, especially under the condition that all items must be different, counting the number of valid orderings usually involves factorial calculations or specific formulas for restricted permutations.

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