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Kristen has six places she needs to go to complete her errands: the bank, grocery store, gas station, hair salon, the dry cleaners, and the library. How many different orders can she choose to complete her errands? (1 point) 36 720 7,776 15,625

Ask by Washington Kelley. in the United States
Mar 04,2025

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Kristen can complete her errands in 720 different orders.

Solución

To determine how many different orders Kristen can choose to complete her errands, we can use the concept of permutations. Since she has six distinct places to visit, the number of different orders is given by the factorial of the number of places. The formula for the factorial of a number \( n \) is: \[ n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \] In this case, \( n = 6 \) (the six places). Therefore, we need to calculate \( 6! \). Let's calculate \( 6! \): \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] Now, we can perform the multiplication step by step: 1. \( 6 \times 5 = 30 \) 2. \( 30 \times 4 = 120 \) 3. \( 120 \times 3 = 360 \) 4. \( 360 \times 2 = 720 \) 5. \( 720 \times 1 = 720 \) Thus, the total number of different orders Kristen can choose to complete her errands is \( 720 \). The correct answer is \( 720 \).

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To find out how many different orders Kristen can complete her errands, we need to calculate the factorial of the number of places she has to go, which is 6. The calculation is 6! (6 factorial), meaning 6 x 5 x 4 x 3 x 2 x 1 = 720. Therefore, the answer is **720** distinct ways for her to complete her errands. Now, let's talk about some fun facts! Did you know that the concept of factorial is foundational in combinatorics and is often involved in calculating permutations and combinations? Factorials pop up in various areas, including probability, statistics, and even in computer science when dealing with algorithms! If Kristen wants to make her errands more efficient, she could consider organizing her route based on proximity. Grouping nearby errands can save time and reduce travel distance. For example, going to the grocery store near the bank can help her minimize backtracking instead of randomly going in any order!

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