Use organized counting strategies

Organized counting is the systematic enumeration of outcomes in finite sample spaces by dividing possibilities into distinct, nonoverlapping cases and applying the addition and multiplication principles, often represented with tree diagrams, tables, or lists. The understanding includes distinguishing ordered from unordered selections and accounting for repetition or restrictions through permutations, combinations, and complementary counting, providing the basis for equally likely probability models; abstract combinatorial methods and generalized infinite cases are not included.

Detailed Explanation: Use organized counting strategies

Organized counting means dividing a problem into clear, nonoverlapping cases, counting each case, and then adding the results. Within one case, use multiplication when choices happen in stages.

Example

How many 4-digit even numbers can be formed from the digits $(0,1,2,3,4,5)$ without repetition?

Step 1: Identify the restrictions

A 4-digit number cannot begin with $0$ .
An even number must end in $(0,2,)$ or $4$ .
No digit may be repeated.

We will organize the count by the possible last digit.

Case 1: The last digit is $0$

The number has the form

\_ \ \_ \ \_ \ 0

First digit: $5$ choices ( $1,2,3,4,5$ )
Second digit: $4$ choices
Third digit: $3$ choices

Using multiplication:

5\cdot 4\cdot 3=60

Case 2: The last digit is $2$ or $4$

Last digit: $2$ choices
First digit: $4$ choices
(It cannot be $0$ or the digit already used.)
Second digit: $4$ choices
Third digit: $3$ choices

Thus,

2\cdot 4\cdot 4\cdot 3=96

Step 3: Add the nonoverlapping cases

A number cannot end in both $0$ and $2$ or $4$ , so the cases do not overlap. Therefore, use addition:

60+96=\boxed{156}

There are $\boxed{156}$ such 4-digit even numbers.

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Probability Fundamental Counting Principle - Scenario Details to Answer

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