Problem 1. World Cup ( 22 points) The 2023 Women's World Cup will feature 32 teams. At the end of the tournament the teams ranked from 1 to 32 . (a) [2 pts.] At the start of the tournament, how many different possible rankings are there of the teams? (b) [2 pts.] Group A consists of New Zealand, Norway, Philippines, and Switzerland. How many different possible orderings are there of these four teams? For example, one possible ordering is New Zealand then Switzerland then Philippines then Norway. (c) [2 pts.] If we consider the positions occupied by the Group A teams in the final rankings, how many different sets of positions could there be? For example, one set is where Group A could occupy positions 4,6,17, and 28 . (Note- you can ignore any nuance to the tournament and just assume that any of the 32 teams can occupy any of the 32 positions, independent of the tournament structure or anything else.) (d) [2 pts.] Now, if we consider the positions occupied by the Group A teams in the final rankings and the teams in those positions, how many different possibilities are there? For example, one possibility is New Zealand could be in position 4, Switzerland could be in position 6, Philippines could be in position 17, and Norway could be in position 28. (Note-you can ignore any nuance to the tournament and just assume that any of the 32 teams can occupy any of the 32 positions, independent of the tournament structure or anything else.) (e) [3 pts.] How many final rankings of the 32 teams have one of the Group A teams in 1st place (as the champion)? Briefly describe how you arrived at your solution. (f) [3 pts.] How many final rankings of the 32 teams have none of the Group A teams in 1st place? Briefly describe how you arrived at your solution. (g) [4 pts.] If we assume that none of the Group A teams play in the championship game (so none of them occupy 1st or 2nd place). then how many final rankings of the 32 teams are there? Describe how you arrived at your solution. (h) [4 pts.] How many final rankings of the 32 teams have New Zealand before Norway before Philippines before Switzerland? One such example would be New Zealand in 4 th, Norway in 8 th, Philippines in 19 th, and Switzerland in 30 th. Describe how you arrived at your solution