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?
(a) To choose four members out of 25 to serve on an executive committee, we can use the combination formula. The number of ways to choose four members is given by:
C(25, 4) = 25! / (4! * (25-4)!) = 25! / (4! * 21!) = (25 * 24 * 23 * 22) / (4 * 3 * 2 * 1) = 12,650 ways.
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 use the permutation formula. The number of ways to arrange four people in different positions is given by:
P(25, 4) = 25! / (25-4)! = 25! / 21! = 25 * 24 * 23 * 22 = 30,240 ways.
Therefore, there are 30,240 ways to choose a president, vice-president, secretary, and treasurer of the club.
(a) To choose four members of the club to serve on an executive committee, we can use the combination formula. The combination formula is given by:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of members in the club and r is the number of members to be chosen for the committee.
In this case, n = 25 (total number of members) and r = 4 (number of members to be chosen).
Using the combination formula, we can calculate the number of ways to choose four members for the executive committee:
C(25, 4) = 25! / (4! * (25 - 4)!)
= 25! / (4! * 21!)
= (25 * 24 * 23 * 22) / (4 * 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 use the permutation formula. The permutation formula is given by:
P(n, r) = n! / (n - r)!
Where n is the total number of members in the club and r is the number of offices to be filled.
In this case, n = 25 (total number of members) and r = 4 (number of offices to be filled).
Using the permutation formula, we can calculate the number of ways to choose a president, vice-president, secretary, and treasurer:
P(25, 4) = 25! / (25 - 4)!
= 25! / 21!
= (25 * 24 * 23 * 22)
= 303,600
Therefore, there are 303,600 ways to choose a president, vice-president, secretary, and treasurer of the club, where no person can hold more than one office.
To solve the problem, we can use the concept of combinations and permutations.
(a) To choose four members of the club to serve on an executive committee, we need to determine the number of combinations. This means that the order in which the members are selected does not matter.
The formula for combinations is given by:
C(n, r) = n! / (r!(n-r)!)
Where:
- n is the total number of members in the club (25 in this case)
- r is the number of members to be chosen for the executive committee (4 in this case)
- "!" denotes the factorial of the number
Using the formula, we can calculate the number of ways to choose four members for the executive committee:
C(25, 4) = 25! / (4!(25-4)!)
Simplifying (using cancellation of the factorials):
C(25, 4) = (25 * 24 * 23 * 22) / (4 * 3 * 2 * 1)
C(25, 4) = 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 need to determine the number of permutations. This means that the order in which the members are selected does matter.
The formula for permutations is given by:
P(n, r) = n! / (n-r)!
Where:
- n is the total number of members in the club (25 in this case)
- r is the number of members to be chosen for different offices (4 in this case)
- "!" denotes the factorial of the number
Using the formula, we can calculate the number of ways to choose a president, vice-president, secretary, and treasurer:
P(25, 4) = 25! / (25-4)!
Simplifying (using cancellation of the factorials):
P(25, 4) = (25 * 24 * 23 * 22)
P(25, 4) = 30,240
Therefore, there are 30,240 ways to choose a president, vice-president, secretary, and treasurer of the club, where no person can hold more than one office.