Textbook Question
True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
3. A combination is an ordered arrangement of objects.
4. Probability
Counting
1:34 minutesProblem 3.4.20
Textbook Question
20. Skating Eight people compete in a short track speed skating race. Assuming that there are no ties, in how many different orders can the skaters finish?
Verified step by step guidance1
Step 1: Recognize that this problem involves permutations, as the order in which the skaters finish matters.
Step 2: Recall the formula for permutations of n distinct items, which is n!. This represents the total number of ways to arrange n items in order.
Step 3: Identify the number of skaters competing in the race, which is 8. Therefore, we need to calculate 8! (8 factorial).
Step 4: Write out the factorial expression for 8!: 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
Step 5: Multiply the terms in the factorial expression to determine the total number of permutations. This will give the number of different orders in which the skaters can finish.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Permutations
Permutations refer to the different ways in which a set of items can be arranged or ordered. In this context, the eight skaters can finish in various sequences, and the total number of unique arrangements is calculated using the factorial of the number of items, denoted as n!. For example, if there are 3 skaters, the number of finishing orders would be 3! = 6.
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Factorial
The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to determine the number of ways to arrange n distinct objects. For instance, 5! = 5 × 4 × 3 × 2 × 1 = 120, which represents the number of ways to arrange 5 skaters.
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Combinatorial Counting
Combinatorial counting involves techniques for counting the arrangements or selections of items in a set. In the case of the skating race, since the order of finish matters and there are no ties, we use permutations to count the possible outcomes. Understanding this concept is crucial for solving problems related to arrangements and sequences in statistics.
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