a realtor uses a lock box to store the keys to a house that is for sale. the access code for the lock box consists of four digits. the first digit cannot be 9 and the last digit must be even. how many different codes are available? (note that 0 is considered an even number.)

the number of different codes available is .
(type a whole number.)

Question

a realtor uses a lock box to store the keys to a house that is for sale. the access code for the lock box consists of four digits. the first digit cannot be 9 and the last digit must be even. how many different codes are available? (note that 0 is considered an even number.)

the number of different codes available is .
(type a whole number.)

the number of different codes available is  .
(type a whole number.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Fundamental Counting Principle", "Combinatorics" ], "new_concepts": [], "current_concepts": [ "Fundamental Counting Principle", "Combinatorics", "Permutations and Combinations" ] } ## Identify the number of choices for each digit position Using the **Fundamental Counting Principle** and **Combinatorics** knowledge points $$ \begin{aligned} &\text{Total digits available: } \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \quad (\text{total of } 10 \text{ digits})\ &\text{Digit 1 choices: } \{0, 1, 2, 3, 4, 5, 6, 7, 8\} \quad (\text{cannot be } 9 \implies 9 \text{ choices})\ &\text{Digit 2 choices: } \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \quad (10 \text{ choices})\ &\text{Digit 3 choices: } \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \quad (10 \text{ choices})\ &\text{Digit 4 choices: } \{0, 2, 4, 6, 8\} \quad (\text{must be even} \implies 5 \text{ choices}) \end{aligned} $$ ## Calculate the total number of available codes Using the **Fundamental Counting Principle** knowledge point $$ \begin{aligned} N &= 9 \times 10 \times 10 \times 5 \ N &= 4500 \end{aligned} $$ The number of different codes available is 4500. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Fundamental Counting Principle" ] }

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Explanation:

Identify the number of choices for each digit position

Calculate the total number of available codes

Answer: