math homework help !!!!?

Question

a) If there are 9 candidates in an election and voters are asked to rank all of the candidates using 9 to 1 for highest to lowest, how many different ranking are possible?

(b) If there are 9 candidates in an election and voters are asked to use a Condorcet system, how many different pairwise comparisons are possible?

Thank you :)

Answer 1

a) there's 9 possible ranks, and 9 candidates.



Imagine the possible ranking of each candidate as slots:



___ ___ ___ ___ ___ ___ ___ ___ ___



How can we fill each slot?



Well the first candidate can get any of 1-9, which is 9 ways. Second candidate can get any of the remaining 8 ranks, 3rd can get any of the remaining 7 ranks, 4th can get any of 6 remaining ranks, all the way to the 9th candidate who can only get the 1 remaining rank .... Since each event is independent, the total number of ranking is given by:



9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 9! = 362880 possible rankings



b) Condorcet system essentially runs one candidate against the other 8.



Start by choosing the first candidate. There are 9 of them, so there are 9 ways to choose the first candidate. In choosing the second of the pair, we only have 8 people left (can't run against yourself), so we can choose the pairs in :



9x8 = 72 ways

Answer 2

A) =9!=362880

B) = 240

Answer 3

wat?!