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 six-digit passwords can be formed from the numbers one to seven if repetition is not allowed?

determine the appropriate counting technique. choose the correct answer below.
a. arrangements with repetition should be used because we make selections from a group of choices.
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 no item may be selected more than once and the order matters.
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:

Determine the appropriate counting technique

Using the Permutations knowledge point:

• The problem asks for the number of different six-digit passwords formed from the numbers one to seven.
• "Repetition is not allowed" means no item may be selected more than once.
• For passwords, the order of the digits matters (e.g., 123456 is different from 654321).
• Therefore, permutations must be used because no item may be selected more than once and the order matters. This corresponds to option E.

Calculate the number of possible passwords

Using the Permutations knowledge point:

• We need to select and arrange \(r = 6\) distinct items from a pool of \(n = 7\) items (the numbers 1 through 7).
• The formula for permutations is:

$$ P(n, r) = \frac{n!}{(n-r)!} $$

• Substituting the values:

$$ P(7, 6) = \frac{7!}{(7-6)!} = \frac{5040}{1} = 5040 $$

Answer:

Question 1

• (A) Arrangements with repetition should be used because we make selections from a group of choices.
• (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 no item may be selected more than once and the order matters.
• (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 possible is <blank>5040</blank>.