Can someone please clarify this for me?
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.
permutations are just combinations where the order matters.
combination: pick a 3-member committee from 10 candidates
permutation: pick a president, vice-president, and secretary.
Sure! Let's go through two different situations to understand the concepts of combinations and permutations.
Situation 1: Selecting a Committee
Suppose you have 10 people, and you want to create a committee with 5 members. In this situation, you are asked to use combinations to determine the number of possible committees.
Combination:
When order doesn't matter, and you are selecting objects from a larger set, we use combinations. In this case, you only care about the group of people you select, not the specific order in which they are chosen.
Explanation:
To calculate the number of combinations, you can use the combination formula: C(n, r) = n!/((n-r)! * r!). In our first situation, we want to select 5 people from the group of 10. Using the combination formula, the number of possible committees can be calculated as C(10, 5) = 10! / (10-5)! * 5! = 10! / 5! * 5! = (10 * 9 * 8 * 7 * 6)/(5 * 4 * 3 * 2 * 1) = 252.
Therefore, there are 252 different possible committees that can be formed from the given group.
Situation 2: Arranging Letters
Now let's consider a different situation where you have the word "APPLE," and you want to find the number of arrangements of the letters. In this case, we will be using permutations.
Permutation:
Permutations come into the picture when order matters. It involves arranging objects from a given set.
Explanation:
To calculate the number of permutations, you can use the formula P(n, r) = n!/(n-r)!. In our second situation, we want to arrange the letters in the word "APPLE," which has 5 letters. Using the permutation formula, the number of possible arrangements is P(5, 5) = 5!/(5-5)! = 5!.
Therefore, there are 5! = 5 * 4 * 3 * 2 * 1 = 120 different possible arrangements of the letters in the word "APPLE."
In summary, combinations are used when the order doesn't matter, and permutations are used when the order matters. The key difference lies in whether the arrangement of objects is significant or not.