Talkshow¶
Problem
A talkshow host invites three politicians, a newswoman and two members of a citizens’ action committee. During the discussion round, the participants will be sitting in a semi-circle with the host in the middle and each participant taken as an individual.
Give an expression which allows to determine the number of possible seating arrangements if no other constraints need to be taken into account.
The station has decided that the newswoman will take a seat next to the host and that to the other side of the host, a politician shall be seated. Determine the number of possible seating arrangements accounting for these constraints.
Solution of part a
If we want to generate all possible seating arrangements, we start with the first seat for which we the choice among six person. For the second seat, five person are left and so on. In total, we obtain
possibilities.
If we indicate the host by H, the politicans by 1, 2, and 3, the newswoman by N and the members of the citizens’ action committee by C and c, we can list all seating arrangements:
Our list indeed comprises \(8\cdot90=720\) different seating arrangements.
Solution of part b
We can attribute the seats by proceeding as follows: The newswoman is placed on one of the two seats next to the host (2 possibilities) and one of the three politicians is seated on the other side of the host (3 possibilities). It remains to place four persons on four seats which, in analogy to our reasoning in part a, yields \(4\cdot3\cdot2\cdot1=24\) possiblities. In total, we obtain \(2\cdot3\cdot24\) different seating arrangements which we can list:
We obtain \(18\cdot8=144\) seating arrangements as expected.
