Question

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 - so 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 most 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 size r 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 to n , 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 roster, 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.

Answer 1