Question

answer the following question using arrangements with repetition, permutations, or combinations. be sure to explain why the particular counting technique applies to the problem.
how many different eight-digit passwords can be formed from the numbers one to eight if repetition is not allowed?

determine the appropriate counting technique. choose the correct answer below.
a. arrangements with repetition should be used because no item may be selected more than once and the order matters.
b. permutations should be used because we make selections from a group of choices.
c. combinations, because no item may be selected more than once and the order does not matter.
d. arrangements with repetition should be used because we make selections from a group of choices.
e. permutations should be used because no item may be selected more than once and the order matters.

calculate how many different passwords are possible.
(type a whole number.)

Explanation:

Identify the counting scenario

Using the Permutations knowledge point

• We need to form an eight-digit password from the numbers 1 to 8.
• Repetition is not allowed (each number is used exactly once).
• The order of the digits in a password matters.

Determine the appropriate technique

Using the Permutations knowledge point

• Since repetition is not allowed and order matters, permutations must be used.
• This matches option E: "Permutations should be used because no item may be selected more than once and the order matters."

Calculate the number of possible passwords

Using the Permutations knowledge point

$$ P(8, 8) = 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 $$

Answer:

Question 1

• A. Arrangements with repetition should be used because no item may be selected more than once and the order matters.
• B. Permutations should be used because we make selections from a group of choices.
• C. Combinations, because no item may be selected more than once and the order does not matter.
• D. Arrangements with repetition should be used because we make selections from a group of choices.
• E. Permutations should be used because no item may be selected more than once and the order matters. (Correct answer)

Question 2

The number of different passwords is <blank>40320</blank>.