Question

determine whether the statement below is true or false. if it is false, rewrite it as a true statement.

the number of different ordered arrangements of \\(n\\) distinct objects is \\(n!\\).

choose the correct answer below.

a. the statement is false. a true statement is, \the number of different ordered arrangements of \\(n\\) distinct objects is \\(n\\).\
b. the statement is false. a true statement is, \the number of different ordered arrangements of \\(n\\) distinct objects is \\((n - 1)!\\).\
c. the statement is true.
d. the statement is false. a true statement is, \the number of different unordered arrangements of \\(n\\) distinct objects is \\(n!\\).\

Explanation:

Analyze the given statement

Using the Combinatorics knowledge point

$$ \text{Statement: "The number of different ordered arrangements of } n \text{ distinct objects is } n!." $$

Evaluate the mathematical validity

Using the Combinatorics knowledge point

$$ P(n, n) = \frac{n!}{(n-n)!} = \frac{n!}{0!} = n! $$

Match with the multiple-choice options

Using the Combinatorics knowledge point

$$ \text{The statement is true. This corresponds to option C.} $$

Answer:

• A. The statement is false. A true statement is, "The number of different ordered arrangements of n distinct objects is n."
• B. The statement is false. A true statement is, "The number of different ordered arrangements of n distinct objects is (n - 1)!."
• C. The statement is true. (Correct answer)
• D. The statement is false. A true statement is, "The number of different unordered arrangements of n distinct objects is n!."