## pancakeslover 2 years ago A president, treasurer, and secretary, all different, are to be chosen, from a club consisting of 10 people. How many different choices of officers are possible if

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1. pancakeslover

a) there are no restrictions? b) A and B will not serve together? c) C and D will serve together or not at all? d) E must be an officer? e) F will only serve if he is the president?

2. kropot72

a) If there are no restrictions the number of choices is given by 10P3.

3. pancakeslover

@kropot72 do you know the answers to the rest?

4. wio

For b), my strategy would be to find how many ways A and B will serve together and subtract that from the total.

5. wio

For c) you are doing a) again but with 8 people to find out how many ways C and D are not at all serving together, then adding to what you got in b) for ways A and B will serve together.

6. wio

d) is very simple, number of way E will serve.

7. wio

e) is number of ways F doesn't serve plus number of ways to assign treasurer and secretary to remaining 9 people

8. wio

Could you explain that better? I'm not sure how you got that.

9. wio

For b) I'm getting: $^{10}P_{3} - (^{3}C_{2} \times ^{8}P_{1})$

10. wio

$$^3C_2$$ - ways to assign A and B a position $$^8P_1$$ - ways to fill the remaining position $$^3C_2\times ^8P_1$$ - ways A and B will serve together $$^{10}P_3-(^3C_2\times ^8P_1)$$ - ways A and B will not serve together

11. wio

c) C and D will serve together or not at all? $$^3C_2\times ^8P_1$$ - ways C and D will server together $$^8P_3$$ - ways neither C nor D will serve $$(^3C_2\times ^8P_1) +^8P_3$$ - ways C and D will server together or not at all

12. wio

d) E must be an officer $$^3C_1$$ - ways E can be an officer $$^9P_2$$ - ways to fill the two remaining positions $$^3C_1 \times ^9P_2$$ - ways E is an officer

13. wio

e) F will only serve if he is the president? $$^1C_1$$ - ways F is president $$^9P_2$$ - ways to fill remaining two seats $$^9P_3$$ - ways F will not server $$(\ ^1C_1\times\ ^9P_2\ )+^9P_3$$ - ways if is president or does not serve

14. pancakeslover

thanks @wio !