A team of at least five volunteers will be selected from seven people who volunteer to work at a food bank.
a) How many different teams are possible?
b) How many different teams are possible if one of them will be the team coordinator?
To answer these questions, we can use the concept of combinations.
a) To find the number of different teams possible without any additional constraints, we can use the combination formula:
C(n, r) = n! / (r! * (n - r)!)
In this case, n represents the total number of people volunteering (7) and r represents the number of people needed for a team (5).
C(7, 5) = 7! / (5! * (7 - 5)!)
= 7! / (5! * 2!)
= (7 * 6 * 5!) / (5! * 2)
= (7 * 6) / 2
= 42 / 2
= 21
So, there are 21 different teams possible without any additional constraints.
b) If one of the volunteers will be the team coordinator, we need to select 4 volunteers from the remaining 6. We can again use the combination formula:
C(n, r) = n! / (r! * (n - r)!)
In this case, n represents the remaining number of people after selecting the team coordinator (6) and r represents the number of people needed for a team (4).
C(6, 4) = 6! / (4! * (6 - 4)!)
= 6! / (4! * 2!)
= (6 * 5 * 4!) / (4! * 2)
= (6 * 5) / 2
= 30 / 2
= 15
So, there are 15 different teams possible if one of them will be the team coordinator.