There will be tryouts for two basketball teams, with 9 players each, from a pool of 22 players. Team A will pick players first. There are two conditions: players A and B will be team captains, and players B and C must be on the same team. To calculate the possible teams: 1) There are 5814 possible combinations for the 8 open spots on Team A, since it picks first. 2) That leaves 11 players for Team B, with 2 spots already filled. There are 330 possible combinations to fill the remaining 7 spots. 3) In total, there are 5814 possible Team A combinations and 330 possible Team B combinations.

1. Basketball Try - Outs

2. The Try- outs If there was a basketball team that's holding tryouts for two teams and there are 22 players to try out where each team supposedly has 9 players each. A B C D E F G H I J K L M N O P Q R S T U V How many possible teams are there if: - Person A & Person B are going to be team captain - 2 People have to be in the same team (Person B and Person C) - Team A Gets to Pick Players First

3. Finding Possible Members for Team A Since team A picks first, let’s try to find the number of possible team member composition for team A… Start by looking at the given…

4. The Sample Space There is a team captain which leaves only 8 open spots to be taken for team A. There is another team Captain for Team B + one person who will always be with him. Out of the 22 players , this leaves us with 19 to choose from…

5. Team A Composition… Looking at it as Y (yes) or N (no), lets try getting 8 players out of the 19 players who are trying out… Y Y Y Y Y Y Y Y N N N N N N N N N N N So we are choosing 8 from 19.. This can be written as… C 19 8

6. Team A Composition = n! (n-r)! * r! Is read as “n” Choose “r” C n r C n r

7. Team A Composition Is read as 19 CHOOSE 8 = 19! (19 - 8)! 8! = 19! 11! 8! C 19 8 C 19 8 C 19 8

8. Team A Composition = 19! 11! 8! = 5814 So there are 5814 possible team combinations for Team A from the 22 players who went for the tryouts C 19 8 C 19 8

9. Team B Composition So now our sample space is reduced to 11 since 8 people were picked from a sample space of 19… This time around we only need 7 more people in the team since there already 2 members that are in it for sure… Y Y Y Y Y Y Y N N N N

10. Team B Composition So we are choosing 7 from 11.. This can be written as… Is read as 11 CHOOSE 7 = 11! (11 - 7)! 7! C 11 7 C 11 7

11. Team B Composition = 11! 4! 7! = 330 So there are 330 possible team combinations for Team B C 11 7 C 11 7

12. CONCLUSION So there are a total of 5814 possible player combinations for team A and there are 330 possible player combinations for team B.