4. Probability
Fundamental Counting Principle
3:18 minutesProblem 4.4.8
Textbook Question
Soccer Shootout In the FIFA Women’s World Cup 2019, a tie at the end of two overtime periods leads to a “shootout” with five kicks taken by each team from the penalty mark. Each kick must be taken by a different player. How many ways can 5 players be selected from the 11 eligible players? For the 5 selected players, how many ways can they be designated as first, second, third, fourth, and fifth?
Verified step by step guidance1
Step 1: Recognize that the problem involves two parts: (1) selecting 5 players from 11 eligible players, and (2) arranging the selected 5 players in a specific order (first, second, third, fourth, and fifth).
Step 2: To calculate the number of ways to select 5 players from 11, use the combination formula: , where is the total number of players (11) and is the number of players to be selected (5).
Step 3: Substitute and into the combination formula to compute the number of ways to select 5 players: .
Step 4: To calculate the number of ways to arrange the 5 selected players in a specific order, use the permutation formula: . Substitute and to compute the number of arrangements: .
Step 5: Multiply the results from Step 3 (number of ways to select 5 players) and Step 4 (number of ways to arrange the 5 players) to find the total number of ways to select and arrange the players.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combinations
Combinations refer to the selection of items from a larger set where the order does not matter. In this context, we need to choose 5 players from a pool of 11 eligible players. The formula for combinations is given by C(n, k) = n! / (k!(n-k)!), where n is the total number of items, k is the number of items to choose, and '!' denotes factorial.
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Permutations
Permutations involve the arrangement of items where the order does matter. After selecting 5 players, we need to determine the different ways to assign them to specific kicking positions (first, second, etc.). The number of permutations of k items from a set of n is calculated using the formula P(n, k) = n! / (n-k)!, which accounts for the order of selection.
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Factorial
Factorial is a mathematical operation that multiplies a number by all positive integers less than it. It is denoted by n! and is crucial in both combinations and permutations calculations. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is essential for calculating the total number of ways to select and arrange players in this soccer shootout scenario.
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