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

Question

anonymous · Answer

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?

kropot72 · Answer

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

anonymous · Answer

@kropot72 do you know the answers to the rest?

anonymous · Answer

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

anonymous · Answer

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.

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

$^3C_2$ - 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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

thanks @wio !