Math Problem Statement
A license plate has 7 characters. Each character can be a capital letter or a digit except for 0. How many license plates are there in which no character appears more than once and the first character is a digit?
Group of answer choices
9⋅P(34,6)
9⋅(35)6
9⋅(34)6
9⋅P(35,6)
Solution
To solve the problem, let's break it down step by step:
Key Points:
- 7 characters: The license plate has 7 characters.
- Character options: Each character can be a capital letter or a digit (excluding 0).
- Total digits = (1 through 9).
- Total letters = (A through Z).
- Total possible characters = .
- First character: The first character must be a digit, and no character can appear more than once.
Step-by-Step Calculation:
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Choose the first character (digit): There are choices for the first character since the first character must be a digit (1 through 9).
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Remaining pool of characters: Once the first character is chosen, there are remaining characters (35 total characters minus the 1 already chosen).
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Remaining 6 characters: These must be chosen from the remaining characters, and no repetitions are allowed. The number of ways to choose and arrange 6 characters from 34 is given by , which represents the number of permutations of 6 items chosen from a pool of 34.
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Total combinations: Multiply the choices for the first character by the number of ways to arrange the remaining 6 characters, .
Final Answer:
This matches the first choice in the options provided.
Follow-up Questions:
- What is the definition of , and how does it differ from combinations ()?
- How would the solution change if the first character could also be the digit 0?
- What is the value of , and how is it calculated?
- If repetition was allowed, how many license plates would be possible?
- How would the solution change if the license plate were limited to exactly 3 letters and 4 digits?
Tip:
Remember, permutations () consider the order of arrangement, whereas combinations () do not. Always identify whether order matters in the problem!
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Permutations
Counting Principles
Formulas
P(n, r) = n! / (n - r)!
Theorems
Permutation theorem
Suitable Grade Level
Grades 9-12
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