Create two different real-world scenarios—one in which you would use permutations and one in which you would use combinations:
Include specific details and an explanation about what makes each situation either a combination or permutation.
Provide sample data and calculate the total number of possible permutations and combinations for both scenarios.
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Scenario 1: Permutations Example - Seating Arrangements
Let's consider a scenario where you are organizing a dinner party with 6 guests and you want to determine the number of ways you can arrange them at a circular table. In this case, we will use permutations since the order of the guests matters.
Explanation:
In permutations, the order or arrangement of items is important. In this scenario, the seating arrangement is a perfect illustration of permutations because each arrangement of the guests will create a different outcome and experience for the dinner party.
Sample Data:
Number of guests (n) = 6
Calculation:
To find the total number of permutations, we can use the formula for permutations:
P(n) = n!
P(6) = 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
Therefore, there are 720 different ways to arrange the 6 guests at the circular table.
Scenario 2: Combinations Example - Lottery Drawing
Let's consider a scenario where you are hosting a lottery drawing with 10 participants, and you want to determine the number of ways you can choose the winners. In this case, we will use combinations as the order of the winners does not matter.
Explanation:
In combinations, the order is not important, and the selection of items is considered as a group. In this scenario, choosing the winners of the lottery is a suitable example of combinations because the order in which each participant is chosen does not affect the result.
Sample Data:
Number of participants (n) = 10
Number of winners to be chosen (r) = 3
Calculation:
To find the total number of combinations, we can use the formula for combinations:
C(n, r) = n! / (r!(n-r)!)
C(10, 3) = 10! / (3!(10-3)!) = 10! / (3! x 7!) = (10 x 9 x 8) / (3 x 2 x 1) = 120
Therefore, there are 120 different ways to choose 3 winners from the 10 participants in the lottery drawing.