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# counting question

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Our club has 10 members, and wishes to pick a president, secretary, treasurer, morale officer, and dog catcher. In how many ways can we choose the officers, if individual members can only hold at most one office?

Jan 8, 2021

#1
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10 ways to pick pres

9 ways to pick secretary

8 ways to pick treasurer

7 ways to pick morale officer

6 ways to pick dog catcher

30240 ways

:D

Jan 8, 2021
#2
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I think you must of meant "..no person can hold 'more than' 1 office..."
This is the case of the number of "different" ways to select 4 items out of 10.
where the order matters (who holds the president office matters) and no repetition is allowed. This is a permutation problem

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so there are 10 choices for president
then after that there are 9 choices for VP
then afte P and VP there are 8 choices for Treasurer
and finally after P, VP and T, there are 7 choices left for Secretary.
So the number of ways to fill those offices with 10 candidates is
$$\frac{10!}{(10-4)!}$$

10*9*8*7*6

30240

Jan 8, 2021
edited by hihihi  Jan 8, 2021