University x needs to send a delegation of athletes to represent the university at a soccer tournament. assume that the soccer roster has n athletes. (a) if the soccer sund expected is of size k, how many different ways can the squad be made up of? (b) in addition, if the soccer squad must also have a captain (selected from the k athletes), how many different "captained" squads are there? note here that selecting the same set of players but assigning a different captain counts as different "captained" squads. (e) if every member of the delegation is to be assigned a jersey numbers range from 1 to k), how many different ways are there of sending a "jerseyed" squad?
(d) the squash team has all its players injured and the coach decides to pick her squad from the soccer roster assume that there are no conflicts among the squash and soccer schedules to any player is free to get selected for both squads (if deemed worthy). the constraint for the squash squad is that there must be at least one player on the squad (and at moet n). again, this squad should have a captain (picked from amongst the squad). use the following 2 equivalent ways to construct the squad and count the number of squash squads possible for each way: • pick a squad of sizer and then select a captain from this lot. any valid squad of size r looks like this. since the size of the squad can range from 1 ton, count the total number of valid squads that may be formed this way. note that any squad picked must have a captain. so, first pick a captain from the roster of size n. how to set up the rest of the squad? well, for every other member (non-captain) of the router, either decide to have them on the squad or not. give an expression that counts the number of squads that can be formed by this process. use the answers from the above two questions to give an identity involving binomial coefficients.