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 31 members is to select five officers (president, vice president, secretary, treasurer, and historian). In how many ways can this be done?
I came up with 155 ways,
Thank you and Good afternoon
It is not combinations because the order matters, it is permutations. (Joe could be either president or vice president for example and those are not the same)
31!/(31-5)!
= 31!/26!
= 31*30*29*28*27
= 20,389,320
thank you Damon I took notes to see how if falls into place.
You are welcome.
This is a good example to diiferentiate between combinations and permutations
Combinations of 31 taken 5 at a time would be
31! /[ 5! (31-5)! ]
That takes into account all the different groups of five you can make, but does not order the five for specific office.
If you now require a different choice for every office within the group of 5, you will get 5! or 120 times as many arrangements
20,389,320 / 120 = 169,911
so I do not know where you got 155
To solve this problem, we need to determine whether we should use permutations or combinations. Permutations are used when the order of selecting the items matters, while combinations are used when the order does not matter.
In this case, the order of selecting the officers does matter because each position (president, vice president, secretary, treasurer, and historian) is different. Therefore, we will use permutations to find the solution.
To calculate the number of ways to select the officers, we will use the formula for permutations of n objects taken r at a time:
P(n, r) = n! / (n - r)!
In this problem, we have n = 31 (the number of members in the club) and r = 5 (the number of officer positions to fill).
Using the permutation formula, we get:
P(31, 5) = 31! / (31 - 5)!
= 31! / 26!
Now, let's calculate the solution:
1. Calculate 31! (factorial of 31)
= 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
2. Calculate 26! (factorial of 26)
= 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
3. Divide the result from step 1 by the result from step 2:
P(31, 5) = 31! / 26!
= (31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) /
(26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)
After simplification, we find:
P(31, 5) = 31 x 30 x 29 x 28 x 27 = 143,102,400
Therefore, there are 143,102,400 ways to select the five officers from the club with 31 members.
So, your initial answer of 155 ways is incorrect. The correct answer is 143,102,400 ways.
I hope this explanation helps! If you have any more questions, feel free to ask.