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:

  1. 7 characters: The license plate has 7 characters.
  2. Character options: Each character can be a capital letter or a digit (excluding 0).
    • Total digits = 99 (1 through 9).
    • Total letters = 2626 (A through Z).
    • Total possible characters = 9+26=359 + 26 = 35.
  3. First character: The first character must be a digit, and no character can appear more than once.

Step-by-Step Calculation:

  1. Choose the first character (digit): There are 99 choices for the first character since the first character must be a digit (1 through 9).

  2. Remaining pool of characters: Once the first character is chosen, there are 3434 remaining characters (35 total characters minus the 1 already chosen).

  3. Remaining 6 characters: These must be chosen from the remaining 3434 characters, and no repetitions are allowed. The number of ways to choose and arrange 6 characters from 34 is given by P(34,6)P(34, 6), which represents the number of permutations of 6 items chosen from a pool of 34.

  4. Total combinations: Multiply the 99 choices for the first character by the number of ways to arrange the remaining 6 characters, P(34,6)P(34, 6).

Total license plates=9P(34,6)\text{Total license plates} = 9 \cdot P(34, 6)

Final Answer:

9P(34,6)\boxed{9 \cdot P(34, 6)}

This matches the first choice in the options provided.


Follow-up Questions:

  1. What is the definition of P(n,r)P(n, r), and how does it differ from combinations (C(n,r)C(n, r))?
  2. How would the solution change if the first character could also be the digit 0?
  3. What is the value of P(34,6)P(34, 6), and how is it calculated?
  4. If repetition was allowed, how many license plates would be possible?
  5. How would the solution change if the license plate were limited to exactly 3 letters and 4 digits?

Tip:

Remember, permutations (P(n,r)P(n, r)) consider the order of arrangement, whereas combinations (C(n,r)C(n, r)) 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