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?

The final answer is 449,064 ways to select five officers from 31 members.
To find the number of ways to select five officers from 31 members, we can use the concept of combinations.
A combination is a selection of items from a larger set where the order of selection doesn't matter.
We'll use the formula for combinations, which is:
C ( n , k ) = k ! ( n − k )! n ! ​
In this case, n = 31 (total members) and k = 5 (officers to be selected).
Substituting the values into the formula:
C ( 31 , 5 ) = 5 ! ( 31 − 5 )! 31 ! ​ = 5 ! 26 ! 31 ! ​
Now, let's calculate:
= 5 × 4 × 3 × 2 × 1 31 × 30 × 29 × 28 × 27 ​ = 5 × 3 27 × 31 × 29 × 28 × 27 ​ = 5 27 × 31 × 29 × 28 ​
Now, let's compute this:
= 449064
So, there are 449 , 064 ways to select five officers from 31 members.

Answered by rubyyira | 2024-06-25

The club can select and arrange five officers from 31 members in 20,350,320 different ways. This is calculated using the permutation formula since the order of selection matters. Therefore, careful calculations show the selections lead to a large number of unique arrangements.
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Answered by rubyyira | 2024-11-04