A committee of 5 people is to be formed from a group of 7 women and 4 men. How many possible committees can be formed if: the committee then must chose a president and a vice president, how many ways could these two positions be chosen?
why is it permutations not combination? I randomly tried 5P2 and got the right answer, but I don't get why it isnt 5C2?
Thanks in advance
For pres and vp
John and Joe is different from
Joe and John
This is why it is a permutation.
on a commitee Joe and John are the same choice as John and Joe which requires a combination.
Your question is not clear at all.
Are we first selecting the committees, then picking the president and vp ?
number of committees
= C(11,5) = 462
to pick a president , then vice president, order does matter, so
11x10 = 110 or P(11,2)
if we are picking the pres and vpres from the 5, then
5x4 = 20 or P(5,2)
To find the number of possible committees that can be formed, we need to consider both the selection of the committee members and the selection of the president and vice president positions.
Let's break down the problem step-by-step:
1. Selection of the committee members:
From a group of 7 women and 4 men, we need to choose 5 people for the committee. Since the order in which the members are selected does not matter, this can be calculated using combinations (nCr). In this case, it is 11C5.
2. Selection of the president and vice president positions:
From the 5 chosen committee members, we need to select 2 people for the president and vice president positions. However, the order of the selection does matter, as the president and vice president are distinct positions. Therefore, we need to use permutations (nPr) to calculate the possibilities. In this case, it is 5P2.
The reason we use permutations instead of combinations in this step is that the president and vice president roles are distinct positions, and the order in which they are selected matters. Using combinations would imply that the order of selection doesn't matter, but in this case, it does.
So, the total number of possible committees with a president and vice president can be calculated as:
11C5 * 5P2 = (11! / (5! * 6!)) * (5! / (3! * 2!)) = 462 * 20 = 9,240.
Thus, there are 9,240 possible committees that can be formed with a president and vice president.
To understand why permutations (5P2) is used instead of combinations (5C2) for choosing a president and vice president from a committee of 5 people, we need to understand the difference between the two concepts.
Permutations and combinations are counting principles used in combinatorics to calculate the number of possible arrangements or selections.
Permutations (nPr) are used when the order of selection matters. In other words, each item is distinct and has a specific position. For example, selecting a president and vice president from a committee requires considering the order in which they are chosen.
Combinations (nCr), on the other hand, are used when the order of selection doesn't matter. In combinations, selecting the same items in a different order is considered as one outcome. For example, if you were selecting two committee members for a task, and the order of their selection didn't matter, you would use combinations.
Now, let's go back to the example. We have a committee of 5 people, and we want to choose a president and a vice president.
To choose the president, we have 5 candidates from the committee of 5 people. Thus, we have 5 options for the presidency.
Once the president is selected, we need to choose a vice president. Since the order matters (the president is distinct from the vice president), we still have 4 remaining candidates to choose from.
To calculate the total number of ways to choose a president and a vice president, we multiply the number of choices for each position. This can be represented as 5P2, which is equal to (5!/3!) = 5x4 = 20.
If we were calculating combinations (5C2), we would be considering the president and vice president as one pair, without distinguishing their positions. In this case, the order of the two positions is not relevant. However, since the positions of president and vice president are different, combinations cannot reflect the distinct roles.
Therefore, for this particular scenario where the order matters, permutations (5P2) is the appropriate method to determine the number of ways to choose a president and a vice president from a committee of 5 people.