A club has 25 members. a) How many ways are there to choose four members of the

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Answered question

2021-10-11

A club has 25 members. a) How many ways are there to choose four members of the club to serve on an executive committee? b) How many ways are there to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office?

Answer & Explanation

Willie

Willie

Skilled2021-10-12Added 95 answers

Definition permulation (order is important):
P(n,r)=n!(nr)!
Definition combination (order is not important):
C(n,r)=(nr)=n!r!(nr)!
with n!=n(n1)...21
Solution
(a) The order of the members does not matter (because each selected member will receive the same position), thus we then need to use the definition of combination.
There are 25 members, of which 4 are selected.
n=25
r=4
Evaluate the definition of a combination:
C(25,4)=25!4!(254)!=25!4!21!=12,650
(b) The order of the members is important (because each selected member will receive a different position), thus we then need to use the definition of permutation.
There are 25 members, of which 4 are selected.
n=25
r=4
Evaluate the definition of a combination:
P(25,4)=25!(254)!=25!21!=25242322=303,600
Result:
(a)12,650 ways
(b)303,600 ways

Jeffrey Jordon

Jeffrey Jordon

Expert2023-04-30Added 2605 answers

a) To choose four members from a group of 25 members, we can use the combination formula:
(254)=25!4!(254)!=25×24×23×224×3×2×1=12,650
Therefore, there are 12,650 ways to choose four members of the club to serve on an executive committee.
b) To choose a president, vice president, secretary, and treasurer of the club, we can first choose the president from the 25 members, then choose the vice president from the remaining 24 members, then choose the secretary from the remaining 23 members, and finally choose the treasurer from the remaining 22 members. We can use the multiplication principle to calculate the total number of ways:
25×24×23×22=303,600
However, since no person can hold more than one office, we have to divide by the number of ways of permuting the four officers. This is given by:
4!=4×3×2×1=24
Therefore, the number of ways to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office, is:
25×24×23×224×3×2×1÷4!=303,60024=12,650
So, there are 12,650 ways to choose the four officers of the club with no person holding more than one office.
Vasquez

Vasquez

Expert2023-04-30Added 669 answers

a) To choose four members from a group of 25 members to serve on the executive committee, we can think of it as placing four indistinguishable balls into 25 distinguishable bins (one for each member). Using stars and bars, we can calculate the number of ways as:
(25+414)=(284)=28×27×26×254×3×2×1=12,650
Therefore, there are 12,650 ways to choose four members of the club to serve on an executive committee.
b) To choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office, we can first choose any four members of the club to fill the positions. Then, we can use the permutation formula to calculate the number of ways of assigning the four positions:
P(4,4)=4×3×2×1=24
However, this includes cases where the same person holds more than one office. Since there are no such cases allowed, we need to subtract the number of such cases. There are four positions and each position can be filled by any of the 25 members, so the number of ways of assigning the four positions to the same person is:
25
Therefore, the number of ways to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office, is:
P(4,4)25=2425=1
This seems like a contradiction, but it arises because there are no valid ways of assigning the positions under the given restrictions. This can be seen by noting that if one person is assigned a position, then there are only 24 members remaining to fill the other three positions, which is not enough. Therefore, there are no valid ways to choose the four officers of the club with no person holding more than one office.
RizerMix

RizerMix

Expert2023-04-30Added 656 answers

Answer:
a) 12,650
b) 3,258,000
Explanation:
a) To find the number of ways to choose four members of the club to serve on an executive committee, we can use the combination formula. The number of ways to choose k items from a set of n items is given by:
(nk)=n!k!(nk)!
In this case, we want to choose 4 members from a set of 25 members. Therefore, the number of ways to choose the committee is:
(254)=25!4!(254)!=25!4!21!=25×24×23×224×3×2×1=12,650
b) To find the number of ways to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office, we can use the permutation formula. The number of ways to choose k items from a set of n items and arrange them in a specific order is given by:
P(n,k)=n!(nk)!
In this case, we want to choose 4 members from a set of 25 members and arrange them in a specific order. Therefore, the number of ways to choose the officers is:
P(25,4)=25!(254)!=25!21!=25×24×23×22=3,258,000
Note that in this case, we cannot use the combination formula because the order in which the officers are chosen matters.

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