Decide whether you would use a permutation, a combination, or neither. Next, write the solution using permutation notation or combination notation, if possible, and, finally, answer the question.
A club with 34 members is to select five officers (president, vice president, secretary, treasurer, and historian). In how many ways can this be done?
33390720
To determine whether to use permutation or combination, we need to consider two factors: order and repetition.
In this scenario, the order matters because each officer position is distinct (president, vice president, etc.). Additionally, once a member is selected for a position, they cannot be chosen again for another position. Therefore, repetition is not allowed.
Since we want to select officers in a specific order without repetition, we will use permutation.
The formula to calculate permutations is:
P(n, r) = n! / (n - r)!
In this case, we have n = 34 (the number of club members) and r = 5 (the number of officers to select).
Now, let's calculate the number of ways to select the officers using permutation notation:
P(34, 5) = 34! / (34 - 5)!
= 34! / 29!
To simplify the calculation, we can cancel out some terms:
= (34 * 33 * 32 * 31 * 30 * 29!) / 29!
The (29!) terms cancel out:
= 34 * 33 * 32 * 31 * 30
Now, we can multiply the remaining terms:
= 171,007,200
Therefore, there are 171,007,200 ways to select the five officers from the club of 34 members.