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pancakeslover
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
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?
a) If there are no restrictions the number of choices is given by 10P3.
@kropot72 do you know the answers to the rest?
For b), my strategy would be to find how many ways A and B will serve together and subtract that from the total.
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.
d) is very simple, number of way E will serve.
e) is number of ways F doesn't serve plus number of ways to assign treasurer and secretary to remaining 9 people
Could you explain that better? I'm not sure how you got that.
For b) I'm getting: \[ ^{10}P_{3} - (^{3}C_{2} \times ^{8}P_{1}) \]
\(^3C_2\) - 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
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
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
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