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?

검증된 단계별 안내

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:

C (n, r) = \frac{n!}{r! (n - r)!}

, where

n

is the total number of players (11) and

r

is the number of players to be selected (5).

Step 3: Substitute

n = 11

and

r = 5

into the combination formula to compute the number of ways to select 5 players:

C (11, 5) = \frac{11!}{5! (11 - 5)!}

Step 4: To calculate the number of ways to arrange the 5 selected players in a specific order, use the permutation formula:

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

. Substitute

n = 5

and

r = 5

to compute the number of arrangements:

P (5, 5) = 5!

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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.