Problem

Specify the number of ways to perform the task described. Give your answers using $P(n, r)$ or $C(n, r)$ notation. The key in recognizing whether a problem involves permutations or combinations is deciding whether order is important.
In order to open your garage door, you must enter a three-digit code on a keypad with numbers $0-9$. Each number may not be used more than once, and the order that you enter the numbers is important.

Answer

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Answer

Final Answer: There are \(\boxed{720}\) ways to enter a three-digit code on a keypad with numbers 0-9, where each number may not be used more than once, and the order that you enter the numbers is important.

Steps

Step 1 :This problem involves permutations because the order in which the numbers are entered is important. We are choosing 3 numbers from a set of 10, and each number can only be used once. This is a permutation problem, which can be represented as \(P(n, r)\), where \(n\) is the total number of items to choose from, and \(r\) is the number of items to choose. In this case, \(n=10\) and \(r=3\).

Step 2 :Calculate the number of permutations using the formula \(P(n, r) = \frac{n!}{(n-r)!}\).

Step 3 :Substitute \(n=10\) and \(r=3\) into the formula to get \(P(10, 3) = \frac{10!}{(10-3)!}\).

Step 4 :Simplify the expression to get the number of permutations as 720.

Step 5 :Final Answer: There are \(\boxed{720}\) ways to enter a three-digit code on a keypad with numbers 0-9, where each number may not be used more than once, and the order that you enter the numbers is important.

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