Offices being held

Question

Using nPr 19 members want to be chosen for 3 officers, Each office to be held by 1 person and no more than 1 person can hold each office. How many ways can those offices be held

Answer 1

There are 19 ways to choose the person for the first office. Then, there are 18 ways to choose the person for the next office. Finally, there are 17 ways to choose the last person. The answer is the product of these 3 numbers.

 

19 * 18 * 17 = 5814.

 

If it was one committee of three people, then the order would not be important. In this case you must divide by the number of permutations of those 3 people which is 3* 2 * 1 or 6. This would result in only 969. That means there are 969 unique 3-person committees that can be created from a group of 19 people. This also means that there are 969 ways to form a committee of 16 people from a group of 19 people. (Each time you create your 3-person committee I can create a 16-person committee by using all of the remaining people from the original 19 people.)