In the following situation, determine whether you are asked to determine the number of permutations or combinations. Then do the calculation.
How many ways can the four offices of chairman, vice chairman, secretary, and treasurer be filled from a club with 28 members?
the four offices are defined and distinct.
e.g, you could not simply interchange the office of the secretary with the office of the chairman
So you have a permutation, since the order matters.
Words such as "committee, select, choose"
usually imply a combination
In this situation, you are asked to determine the number of ways the four offices can be filled from a club with 28 members. To solve this, we need to determine whether we are looking for permutations or combinations.
Permutations are used when the order of the elements matters, while combinations are used when the order does not matter.
In this case, the order of filling the offices does matter since the positions of chairman, vice chairman, secretary, and treasurer are distinct and have different responsibilities. Therefore, we need to find the number of permutations.
To calculate the number of permutations, we can use the formula:
nPr = n! / (n - r)!
where n is the total number of items and r is the number of items chosen.
In this scenario, we need to find the number of ways to fill the four offices (r = 4) from a club with 28 members (n = 28).
Using the formula, we have:
nPr = 28! / (28 - 4)!
Simplifying this expression give us:
nPr = 28! / 24!
Now, we can calculate the number of permutations by evaluating this expression.