Create two different situations: one in which you use combinations and one in which you use permutations. Include specific details and an explanation about what makes each situation either a combination or permutation.
Google "combinations and permutations."
Situation 1: Combinations
Let's consider a scenario where you are selecting a team for a school project. There are 10 students in your class, and you need to choose a group of 5 students to form the team.
To determine the number of different team combinations, we use combinations. This is because the order in which the students are selected does not matter, and once a student is selected, they cannot be selected again.
Using the combination formula, denoted as nCr (read as "n choose r"), we can calculate the number of ways to choose the team of 5 students. In this case, n (the total number of students) is 10, and r (the number of students to be chosen) is 5.
The formula for combinations is:
nCr = n! / (r!(n-r)!)
So, for our scenario:
10C5 = 10! / (5!(10-5)!)
= 10! / (5! * 5!)
= (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1)
= 252
Hence, there are 252 different team combinations possible.
Situation 2: Permutations
Let's imagine you are arranging a seating plan for a dinner party with 6 guests. There are 6 chairs around the dining table, and you want to determine the number of ways to seat the guests.
In this situation, the order in which the guests are seated does matter. Additionally, once a guest is seated, the seat is no longer available for another guest. Therefore, we use permutations to calculate the number of seat arrangements.
The formula for permutations, denoted as nPr (read as "n permute r"), is used to determine the number of ways to arrange objects when order matters.
In our case, n (the total number of guests) is 6, and r (the number of seats to be filled) is also 6. So, the formula becomes:
6P6 = 6! / (6-6)!
= 6! / 0!
= 6!
Since the factorial of 0 is defined as 1, the calculation is simplified to:
6P6 = 6!
Therefore, there are 6! (6 factorial) or 720 different seat arrangements possible.
In summary, combinations are used when the order does not matter, and the items being chosen are distinct (i.e., once selected, they are not selectable again). On the other hand, permutations are used when the order matters and the items being arranged are distinct.