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

a) To choose four members from a group of 25 members, we can use the combination formula:
$\left(\genfrac{}{}{0ex}{}{25}{4}\right)=\frac{25!}{4!(25-4)!}=\frac{25\times 24\times 23\times 22}{4\times 3\times 2\times 1}=12,650$
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:
$25\times 24\times 23\times 22=303,600$
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:
$4!=4\times 3\times 2\times 1=24$
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:
$\frac{25\times 24\times 23\times 22}{4\times 3\times 2\times 1}÷4!=\frac{303,600}{24}=12,650$
So, there are 12,650 ways to choose the four officers of the club with no person holding more than one office.

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:
$\left(\genfrac{}{}{0ex}{}{25+4-1}{4}\right)=\left(\genfrac{}{}{0ex}{}{28}{4}\right)=\frac{28\times 27\times 26\times 25}{4\times 3\times 2\times 1}=12,650$
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:
$P(4,4)=4\times 3\times 2\times 1=24$
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:
$P(4,4)-25=24-25=-1$
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.

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:
$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$
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:
$\left(\genfrac{}{}{0ex}{}{25}{4}\right)=\frac{25!}{4!(25-4)!}=\frac{25!}{4!21!}=\frac{25\times 24\times 23\times 22}{4\times 3\times 2\times 1}=\boxed{12,650}$
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:
$P(n,k)=\frac{n!}{(n-k)!}$
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:
$P(25,4)=\frac{25!}{(25-4)!}=\frac{25!}{21!}=25\times 24\times 23\times 22=\boxed{3,258,000}$
Note that in this case, we cannot use the combination formula because the order in which the officers are chosen matters.