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?

Correct if me if I am wrong. Should a) be 7C5 + 7C6 + 7C7 = 29
b) be 7P1(6C4 + 6C5 + 6C6)

Let's first clarify the concepts of combinations and permutations, as they are crucial to solving these questions.

Combinations (denoted as nCr) represent the number of ways to choose a certain number of items from a larger set without considering the order of selection. In combinations, the order does not matter.

Permutations (denoted as nPr) represent the number of ways to arrange or select items from a larger set while considering the order of selection. In permutations, the order does matter.

a) To determine the number of different teams possible without a specific coordinator, we need to calculate the combinations of selecting five volunteers from the seven who have volunteered.

The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the desired number of items to be selected.

In this case, n = 7 (the total number of volunteers) and r = 5 (the desired number of volunteers in each team).

Therefore, the number of different teams possible is 7C5, which can be calculated as:
7C5 = 7! / (5!(7-5)!) = 7! / (5! * 2!) = (7 * 6 * 5!) / (5! * 2 * 1) = (7 * 6) / (2 * 1) = 21.

So there are 21 different teams possible without a specific coordinator.

b) Now, let's consider the case where one of the volunteers will be the team coordinator. In this case, we need to choose four volunteers from the remaining six (since one has already been chosen as the coordinator).

To calculate the number of different teams possible, we need to multiply the number of permutations of selecting four volunteers from the remaining six (6P4) by the number of ways to choose the coordinator (7P1).

The formula for permutations is nPr = n! / (n-r)!, where n is the total number of items and r is the desired number of items to be selected.

Let's calculate it step by step:

First, we calculate the number of ways to choose the coordinator: 7P1 = 7! / (7-1)! = 7.

Next, we calculate the number of permutations for selecting four volunteers from the remaining six: 6P4 = 6! / (6-4)! = 6! / 2! = 6 * 5 * 4 * 3 = 360.

Finally, we multiply the two results: 7 * 360 = 2520.

So, there are 2520 different teams possible with a specific coordinator chosen.