A club with 33 members is to select five officers (president, vice president, secretary, treasurer, and historian). In how many ways can this be done?
The five posts are distinct (i.e. no duplicate posts).
So the order of selection of these posts is important => solution is a permutation.
Selecting r objects from n distinct objects, where order is important is given by P(n,r) where
P(n,r)=n!/(n-r)!
n! = n factorial
For the given example, n=33, r=5.
Can you take it from here?
To find the number of ways to select five officers from a club with 33 members, we can use the concept of combinations.
The number of ways to select a subset of k objects from a set of n objects is given by the formula C(n, k), also known as "n choose k." In this case, we need to find C(33, 5).
The formula for C(n, k) is:
C(n, k) = n! / (k!(n - k)!)
where "!" denotes a factorial, which means multiplying a number by all the positive integers less than it down to 1.
Plugging in the values for this problem, we have:
C(33, 5) = 33! / (5!(33 - 5)!)
Simplifying this equation, we get:
C(33, 5) = 33! / (5! * 28!)
Now let's break it down further:
33! = 33 * 32 * 31 * ... * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
28! = 28 * 27 * ... * 3 * 2 * 1
Now we can substitute these simplifications back into the equation:
C(33, 5) = (33 * 32 * 31 * ... * 3 * 2 * 1) / ((5 * 4 * 3 * 2 * 1) * (28 * 27 * ... * 3 * 2 * 1))
After canceling the common terms, we are left with:
C(33, 5) = (33 * 32 * 31 * 30 * 29) / (5 * 4 * 3 * 2 * 1)
Now we can calculate the value using a calculator or by simplifying further:
C(33, 5) = 33,633,600 / 120
C(33, 5) = 280,168
Therefore, there are 280,168 ways to choose five officers from a club with 33 members.