Question

answer the following question using the appropriate counting technique, which may be either arrangements with repetition, permutations, or combinations. be sure to explain why this counting technique applies to the problem.

how many license plates can be made of the form xxxx-yy, where x is a letter of the alphabet and y is a numeral 0-9?

what counting technique should be used to make this calculation?

a. arrangements with repetitions because there are \\(r\\) selections from a group of \\(n\\) choices and choices can be repeated.
b. arrangements with repetitions because the selections come from a single group of items, and the order of the arrangement matters.
c. combinations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement does not matter.
d. permutations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement matters.

Explanation:

Analyze the license plate structure

Using the Arrangements with Repetitions knowledge point

$$ \text{Format: } XXXX\text{-}YY $$

$$ X \in \{\text{A, B, C, ..., Z}\} \quad (26 \text{ choices}) $$

$$ Y \in \{0, 1, 2, ..., 9\} \quad (10 \text{ choices}) $$

Determine the counting characteristics

Using the Arrangements with Repetitions knowledge point

• The order of the letters and numerals matters.
• The letters and numerals can be repeated (e.g., AAAA-11 is valid).
• Selections are made with replacement from the respective sets.

Match with the correct definition

Using the Arrangements with Repetitions knowledge point

• Option A states: "Arrangements with repetitions because there are \(r\) selections from a group of \(n\) choices and choices can be repeated."
• This matches the definition of arrangements with repetition where order matters and elements can be repeated.

Answer:

• (A) Arrangements with repetitions because there are r selections from a group of n choices and choices can be repeated. (Correct answer)
• (B) Arrangements with repetitions because the selections come from a single group of items, and the order of the arrangement matters.
• (C) Combinations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement does not matter.
• (D) Permutations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement matters.